Category Theory: Applications to Algebra, Logic and Topology. Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 962 Category Theory Applications to Algebra, Logic and T...

Author: K.H. Kamps | D. Pumplün | W. Tholen (Editors)

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

962 Category Theory Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981

Edited by K.H. Kamps, D. Pumplen, and W. Tholen

Springer-Verlag Berlin Heidelberg New York 1982

Editors

Klaus Heiner Kamps Dieter Pumpl0n Walter Tholen Fachbereich Mathematik und Informatik Fernuniversit~t - Gesamthochschule L0tzowstr. 125, 5800 Hagen Federal Republic of Germany

AMS Subject Classifications (1980): 18-06, 03D, 05C, 06D, 08A, 13C, 13E, 16A, 18A, 18B, 18C, 18D, 18F, 18G, 20L, 26E, 46A, 46B, 46G, 46M, 54B, 54D, 54E, 55F, 55N, 55P, 57M ISBN 3-540-11961-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-11961-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

During

the last stages

this volume

the editors

death of our colleague Her personality remembered

of the p r e p a r a t i o n learnt Graciela

of the tragic Salicrup.

and her work will

by all of us.

of

always

be

PREFACE

The I n t e r n a t i o n a l to Algebra, 1981;

C o n f e r e n c e on C a t e g o r y T h e o r y - A p p l i c a t i o n s

Logic and T o p o l o g y - was h e l d in G u m m e r s b a c h , J u l y

it was a t t e n d e d by 93 m a t h e m a t i c i a n s

6-10,

from 19 d i f f e r e n t coun-

tries. Financial

support

for this c o n f e r e n c e was p r o v i d e d by a grant of

the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t a d d i t i o n a l means of the M i n i s t e r Landes N o r d r h e i n - W e s t f a l e n . their sincere thanks

(grant no.

4851/140/80)

fur W i s s e n s c h a f t

and by

und F o r s c h u n g des

The o r g a n i z e r s w o u l d like to e x p r e s s

for this f i n a n c i a l

assistance,

without which

this c o n f e r e n c e w o u l d not have been possible. The c o n f e r e n c e had been d i v i d e d into three sections: c a t e g o r y theory,

c a t e g o r y theory and logic,

c a t e g o r y theory to analysis,

General

and a p p l i c a t i o n s

t o p o l o g y and c o m p u t e r science.

very m u c h a p p r e c i a t e d by the o r g a n i z e r s

of It was

that John Gray agreed to

be c h a i r m a n of this c o n f e r e n c e

and special thanks are due to him

for his e s s e n t i a l

to its success.

contribution

also very g r a t e f u l

to Horst H e r r l i c h

the section on a p p l i c a t i o n s logy and c o m p u t e r

The o r g a n i z e r s w o u l d

ning of the c o n f e r e n c e

of c a t e g o r y theory to analysis,

like to express Prof.

Peters,

its p r e p a r a t i o n e s s e n t i a l

D u r i n g the c o n f e r e n c e

and e f f e c t i v e help was given and this help has

b e e n g r a t e f u l l y a c k n o w l e d g e d by the o r g a n i z e r s . from the u n i v e r s i t y a d m i n i s t r a t i o n

Thanks

E s p e c i a l l y Mr.

should be m e n t i o n e d

for

for this conference.

are due to the F a c h b e r e i c h M a t h e m a t i k und I n f o r m a t i k of

the F e r n u n i v e r s i t ~ t Many c o l l e a g u e s its preparation. Mrs.

for his ope-

and for the w e l c o m e he e x t e n d e d to the par-

by the a d m i n i s t r a t i o n of the F e r n u n i v e r s i t ~ t ,

Bl0mel

topo-

t h e i r thanks to the Rektor

Dr. Dr. b . c . O .

t i c i p a n t s on b e h a l f of the F e r n u n i v e r s i t ~ t .

his e n g a g e m e n t

are

science.

of the F e r n u n i v e r s i t ~ t ,

and d u r i n g

The o r g a n i z e r s

for his help as c h a i r m a n for

for s u p p o r t i n g this c o n f e r e n c e

in every respect.

a d v i s e d and a s s i s t e d us d u r i n g the c o n f e r e n c e We w o u l d

I. M U l l e r and Mrs.

like e s p e c i a l l y to

and

thank the s e c r e t a r i e s

K. T o p p for their m o s t e f f i c i e n t work.

VJ

Last, Dr.

b u t by no m e a n s

G. Greve,

T. MUller,

Dr. W.

least, Sydow,

all m e m b e r s

of the F e r n u n i v e r s i t ~ t forts

that

there w e r e

ference

and they

ference

feel

This of this

this

our s i n c e r e

Klaus

Heiner

to e x p r e s s

D. BrUmmer,

of the F a c h b e r e i c h for their

Dr.

und

It is due

difficulties

to m a k e

our t h a n k s

B. H o f f m a n n

Mathematik

engagement.

no o r g a n i z a t i o n a l

did t h e i r best

of S p r i n g e r

conference.

series.

Dr.

like

to

and Dr.

Informatik

to t h e i r

during

the p a r t i c i p a n t s

ef-

the con-

of the con-

at ease.

volume

ger L e c t u r e

we w o u l d

Notes All

Lecture

We w o u l d

like

in M a t h e m a t i c s contributions

thanks

Kamps

Notes

Dieter

volume

referees

PumplUn

the p r o c e e d i n g s

the e d i t o r s

for a c c e p t i n g

to this

go to all the

constitutes

to t ha n k

of the S p r i n -

the p r o c e e d i n g s have

been

for

refereed

for their work.

Walter

Tholen

and

PARTICIPANTS

M. A d e l m a n C. A n g h e l H. B a r g e n d a M. B a r r J.M. Beck H.L. B e n t l e y G.J. B i r d R. B S r g e r D. B o u r n H. B r a n d e n b u r g R.D. B r a n d t R. B r o w n C. C a s s i d y Y. Diers G. D u b r u l e A. Duma J.W. D u s k i n R. D y c k h o f f A. Frei P. F r e y d A. F r ~ l i c h e r J.W. Gray C. G r e i t h e r G. G r e v e R. G u i t a r t R. H a r t i n g M. H ~ b e r t H. H e r r l i c h P.J. H i g g i n s M. H ~ p p n e r B. H o f f m a n n R.-E. H o f f m a n n M. H u ~ e k J. Isbell B. Jay P.T. J o h n s t o n e K.H. K a m p s G.M. K e l l y H. K l e i s l i A. K o c k J. L a m b e k H. L i n d n e r F.E.J. L i n t o n H. L o r d R.B. LGs chow J. M a c D o n a l d S. M a c L a n e

L. M ~ r k i G Maury A MSbus T MGller C J. M u l v e y A Mysior R Nakagawa G Naud~ L.D. Nel S.B. N i e f i e l d A. O b t u ~ o w i c z B. P a r e i g i s J. P e n o n M. P f e n d e r A.M. Pitts H.-E. P o r s t T. P o r t e r A. P u l t r D. P u m p l G n R. R e i t e r G. R i c h t e r R. R o s e b r u g h J. R o s i c k ~ G. S a l i c r u p B.M. S c h e i n D. S c h u m a c h e r F. Schwarz Z. S e m a d e n i T. S p i r c u G.E. S t r e c k e r R. S t r e e t T. S w i r s z c z W. S y d o w M. T h i ~ b a u d T. T h o d e W. T h o l e n V.V. T o p e n t c h a r o v V. T r n k o v ~ K. U l b r i c h R.F.C. W a l t e r s H. W e b e r p a l s S. W e c k R. W i e g a n d t A. W i w e g e r R.J. W o o d O. Zurth

AUTHORS'

H.L.

Bentley

ADDRESSES

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of T o l e d o Toledo, Ohio 43605 U.S.A.

R. Betti

Istituto Matematico U n i v e r s i t ~ di M i l a n o Via Saldini 50, M i l a n o Italy

F. B o r c e u x

U n i v e r s i t ~ C a t h o l i q u e de Louvain 1348-Louvain-La-Neuve Belgium

D. B o u r n

U n i v e r s i t ~ de P i c a r d i e U.E.R. de M a t h ~ m a t i q u e s 33, rue St Leu 80039 A m i e n s France

H. B r a n d e n b u r g

I n s t i t u t fur M a t h e m a t i k I Freie U n i v e r s i t ~ t B e r l i n A r n i m a l l e e 2-6 10OO B e r l i n 33 Fed. Rep. of G e r m a n y

R. B r o w n

School of M a t h e m a t i c s and Computer Science U n i v e r s i t y C o l l e g e of N o r t h Wales Bangor, G w y n e d d LL57 2UW U.K.

Y. Diers

D ~ p a r t e m e n t de ~ ""a t h e"m a t l"q u e s U.E.R. des S c i e n c e s U n i v e r s i t ~ de V a l e n c i e n n e s 59326 V a l e n c i e n n e s France

A. Frei

Mathematics Department U n i v e r s i t y of B r i t i s h C o l u m b i a V a n c o u v e r , B.C. C a n a d a V6T IY4

A. F r 6 1 i c h e r

S e c t i o n de M a t h ~ m a t i q u e s U n i v e r s i t ~ de G e n ~ v e 2-4, rue du Li~vre 1 2 1 1 G e n ~ v e 24 Switzerland

IX

J.W. Gray

Department of Mathematics University of Illinois Urbana, Ill. 61801 U.S.A.

G. Greve

Fachbereich Mathematik und Informatik Fernuniversit~t 5800 Hagen Fed. Rep. of Germany

P.J. Higgins

Department of Mathematics University of Durham Science Laboratories South Road Durham DHI 3LE U.K.

R.-E. Hoffmann

Fachbereich Mathematik Universit~t Bremen 2800 Bremen 33 Fed. Rep. of Germany

M. H~ppner

Fachbereich MathematikInformatik Universit~t-GesamthochschulePaderborn 4790 Paderborn Fed. ReD. of Germany

M. Husek

Matematick~ Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia

S. Kaijser

Uppsala University Uppsala Sweden

J. Lambek

Department of Mathematics McGill University 805 Sherbrooke St. West Montreal, PQ Canada H3A 2K6

J. MacDonald

Mathematics Department University of British Columbia Vancouver, B.C. Canada V6T IY4

L. M~rki

Mathematical Institute Hungarian Academy of Sciences Re~itanoda u. 13-15 1053 Budapest Hungary

A. M e l t o n

D e p a r t m e n t of C o m p u t e r S c i e n c e W i c h i t a State U n i v e r s i t y W i c h i t a , K a n s a s 67208 U.S.A.

A. M y s i o r

I n s t i t u t e of M a t h e m a t i c s U n i v e r s i t y of G d a n s k 80952 G d a n s k Poland

L.D. Nel

D e p a r t m e n t of M a t h e m a t i c s Carleton University Ottawa, O n t a r i o C a n a d a KIS 5B6

S.B. N i e f i e l d

Union C o l l e g e S c h e n e c t a d y , N.Y. U.S.A.

J.W.

Pelletier

12308

F a c u l t y of Arts York U n i v e r s i t y 4700 Keele Street Downsview, O n t a r i o C a n a d a M3J IP3

M. P f e n d e r

M A 7-I Technische Universit~t Berlin Str. des 17. Juni 135 1OOO B e r l i n Fed. Rep. of G e r m a n y

H.-E.

Fachbereich Mathemaik Universit~t Bremen 2800 B r e m e n 33 Fed. Rep. of G e r m a n y

Porst

T. P o r t e r

School of M a t h e m a t i c s and C o m p u t e r Science U n i v e r s i t y C o l l e g e of North Wales Bangor, G w y n e d d LL57 2UW U.K.

A. P u l t r

Matematick~ Ustav University Karlova S o k o l o v s k ~ 83 18600 P r a h a Czechoslovakia

R. Reiter

Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 10OO B e r l i n Fed. Rep. of G e r m a n y

Xl

G. R i c h t e r

F a k u l t ~ t fHr M a t h e m a t i k Universit~t Bielefeld U n i v e r s i t ~ t s s t r . 25 4800 B i e l e f e l d I Fed. Rep. of G e r m a n y

M. S a r t o r i u s

Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 1000 B e r l i n Fed. Rep. of G e r m a n y

T. S p i r c u

National Institute for S c i e n t i f i c and T e c h n i c a l Creation D e p a r t m e n t of M a t h e m a t i c s Bdul P~cii 220 79622 B u c h a r e s t Romania

A. Stone

Mathematics Department UC Davis Davis, C a l i f o r n i a U.S.A.

G.E.

D e p a r t m e n t of M a t h e m a t i c s K a n s a s State U n i v e r s i t y M a n h a t t a n , Kansas 66506 U.S.A.

Strecker

R. Street

S c h o o l of M a t h e m a t i c s and Physics Macquarie University N o r t h Ryde, N.S.W. 2113 Australia

W. S y d o w

F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y

J. T a y l o r

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of D u r h a m Science Laboratories South Road D u r h a m DHI 3LE U.K.

W. T h o l e n

F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y

Xil

V. Trnkov~

Matematick9 Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia

R.F.C. Walters

Department of Pure Mathematics University of Sydney N.S.W. 2006 Australia

R. Wiegandt

Mathematical Institute Hungarian Academy of Sciences Re~itanoda u° 13-15 1053 Budapest Hungary

A. Wiweger

Institute of Mathematics Polish Academy of Sciences ~niadeckich 8 00-950 Warszawa Poland

CONTENTS

H.L.

Bentley A note

R. Betti The

on the h o m o l o g y

and R.F.C. symmetry

of r e g u l a r

nearness

spaces

Walters

of the C a u c h y - c o m p l e t i o n

of a c a t e g o r y

F. B o r c e u x On a l g e b r a i c

localizations

13

D. B o u r n A canonical to c o h e r e n t

H. B r a n d e n b u r g A remark

R. B r o w n

Y.

on c a r t e s i a n

An a p p l i c a t i o n 23

closedness

33

Higgins

complexes

and n o n - a b e l i a n

extensions

39

Diers Un c r i t ~ r e de r e p r ~ s e n t a b i l i t ~ de f a i s c e a u x

A.

limits.

and M. H u ~ e k

and P.J.

Crossed

a c t i o n on i n d e x e d homotopy

par

sections

continues 51

Frei Kan e x t e n s i o n s

and s y s t e m s

of i m p r i m i t i v i t y

62

A. F r ~ l i c h e r Smooth

J.W.

structures

69

Gray Enriched

algebras,

spectra

and h o m o t o p y

limits

82

G. Greve General construction topological, uniform

of m o n o i d a l c l o s e d s t r u c t u r e s and n e a r n e s s spaces

in 100

XlV

P.J.

H i g g i n s and J. T a y l o r The f u n d a m e n t a l g r o u p o i d and the h o m o t o p y c r o s s e d c o m p l e x of an o r b i t space

115

R.-E. H o f f m a n n Minimal

topological

completion

of ~ B a n I -->

~Vec

123

M. H ~ p p n e r On the freeness of W h i t e h e a d - d i a g r a m s

133

M. Hu~ek Applications

of c a t e g o r y t h e o r y to u n i f o r m structures

138

S. K a i j s e r and W. P e l l e t i e r A categorical

framework

for i n t e r p o l a t i o n t h e o r y

145

J. L a m b e k 153

T o p o s e s are m o n a d i c over c a t e g o r i e s

J. M a c D o n a l d and A. Stone Essentially monadic adjunctions

167

J. M a c D o n a l d and W. T h o l e n Decomposition factors

of m o r p h i s m s

into i n f i n i t e l y m a n y 175

L. M&rki and R. W i e g a n d t 190

R e m a r k s on r a d i c a l s in c a t e g o r i e s

A. M e l t o n and G.E.

Strecker

On the s t r u c t u r e of f a c t o r i z a t i o n

structures

197

A. M y s i o r A remark on s c a t t e r e d spaces

209

L.D. Nel B o r n o l o g i c a l L 1 - f u n c t o r s as Kan e x t e n s i o n s Riesz-like representations

and 213

XV

S.B. Niefield Exactness and projectivity

221

M. Pfender, R. Reiter, and M. Sartorius Constructive arithmetics

228

H.-E. Porst Adjoint diagonals for topological completions

237

T. Porter Internal categories and crossed modules

249

A. Pultr Subdirect irreducibility and congruences.

256

G. Richter Algebraic categories of topological spaces

263

T. Spircu Extensions of a theorem of P. Gabriel

272

R. Street Characterization of bicategories of stacks

282

W. Sydow On hom-functors and tensor products of topological vector spaces

292

V. Trnkov~ Unnatural isomorphisms of products in a category

302

A. Wiweger Categories of kits, coloured graphs, and games

312

A Note on the Homology of Regular Nearness Spaces H. L. Bentley Abstract:

I t is shown that the homology and cohomology groups of a regular near-

ness space can be defined by means of a v a r i a t i o n on the ~ech method, which uses nerves of uniform covers:

the v a r i a t i o n involves associating with each uniform

cover, not the nerve, but a complex, called the vein, defined by means of nearness In a recent paper, the author showed that the ~ech homology and cohomology groups (: Vietoris homology and Alexander cohomology groups) of merotopic and nearness spaces s a t i s f y , in a variant form, a l l the axioms of Eilenberg-Steenrod. For d e f i n i t i o n s of these groups and f o r h i s t o r i c a l information, the reader is referred to that paper [ I ] .

We are interested here in regular nearness spaces

(for the d e f i n i t i o n , see Herrlich [5]) and in the p o s s i b i l i t y of using what i s , formally, a d i f f e r e n t d e f i n i t i o n of the homology and cohomology groups, but a d e f i n i t i o n which we prove gives r i s e to the usual ~ech groups. By a pair (X, Y) of nearness spaces we mean a nearness space X together with a nearness subspace Y of X. where ~ I

A uniform cover of (X, Y) is a pair 0~= ( C ~ I , O ~ 2)

is a uniform cover of X, ~2C-011 , and ~ 2 L ) { X - Y} is a uniform

cover of Y.

~

The nerve K(01) of a uniform cover ~ =

( C)~I , (~I 2) of (X, Y) is a pair of

s i m p l i c i a l complexes K ( ~ I ) = ( K I ( O I ) , K 2 ( ~ ) ) . elements of ~ I ; ~-~C~ ~

•

a simplex of K I ( ~ I )

The vertices of K I ( ( ~ ) are the

is a f i n i t e subset C~of ~ I such that

The vertices of K2(OI ) are the elements of ~ 2 ;

K2((~) is a f i n i t e subset ( ~ o f

OI 2 such that Y ~ ~ C ~

a simplex of

~ ~o

Recall that a collection C)jL of subsets of a nearness space X is said to be near in X i f f o r each uniform cover ~ of X there exists C e ~ G e C~ ,

C F'~G #

~ .

such that for a l l

Recall also that i f Y is a nearness subspace of X then

a collection ( ~ o f subsets of Y is near in Y i f and only i f ~ is near in X. Now we are ready to make our main d e f i n i t i o n ; i t is a v a r i a t i o n on the d e f i n i t i o n of the nerve.

The vein J(01 ) of a uniform cover C~: ( (-~I' 012) of (X, Y) is a pair of simplicial complexes J( ~ ) = (Jl ( C)I ), J2 ( 0 ] ) ) . the elements of ~ l ;

The vertices of Jl ( 0 1 ) are

a simplex of Jl ( ~ ) is a finite subset (~of 011 such that

C~ is near in X. The vertices of J2 ( 01 ) are the elements of (#~2; a simplex of J2 (

) is a finite subset ~ o f

C)I2 such that

If ~ = ( (Y~I' 01 2) and ~ =

( ~l'

C~A {Y} is near in Y.

J~2 ) are uniform covers of the pair

(X, Y) of nearness spaces then we say that (#~ is a refinement of ~_~ i f refinement of ~Fl and 012 is a refinement of ~2"

('~l is a

Under this relation of re-

finement, the set of all uniform covers of a pair of nearness spaces becomes a directed set. Thus, there is a spectrum of complexes K(OI )

~. K( ~LF )

J( Ol )

"~J(~)

and of complexes

for ~ a refinement of ~J.

From these spectra there arise two spectra of homology

groups and two of cohomology groups. From now on, let G be a fixed abelian group. G will be the coefficient group of our homology and cohomology theories but explicit denotation of G will be suppressed. The direct spectrum of cohomology groups oC~

: Hn(K( ~J ))

~.Hn(K( 01 )) V

has for its limit group the n-dimensional Cech cohomology group of (X, Y) which we will denote by ~n(x, Y).

/~

The inverse spectrum of homology groups

: Hn(K(OI ))

!>Hn(K(~T ))

has for its limit group the n-dimensional ~ech homology group of (X, Y) which we will denote by ~n(X, Y). The direct spectrum of cohomology groups ~

: Hn(j( ~

))

> Hn(j( C)] ))

has for its l i m i t group the n-dimensional vascular cohomology group, of (X, Y) which we will denote by Hn(x, Y).

The inverse spectrum of homology groups

3

OI ~Zy

: Hn(J( 01 ))

-Hn(J(~

))

has for its limit group the n-dimensional vascular homology group of (X, Y) which we will denote by Hn(X, Y). We are now ready for the statement of our main result. Theorem.

I f (X, Y) is a pair of regular nearness spaces then the ~ech and

vascular homology, and cohomology, groups coincide, i.e. Hn(X, Y) = ~n(X, Y)

and

~n(x, Y) = ~n(x, Y) for a l l n. Proof: dual.

We give a proof only for the homology groups; the proof for cohomology is With each collection~u% of subsets of X, we associate the collection ~*

=

{E C X I for some D e ~

Of course, as usual we are using the notation uniform cover of X. write

E < D to mean that {D, X - E} is a

For each uniform cover ~ = ( ~ I '

~ * = ( (~ I * '

012) of (X, Y) we w i l l

~ 2 * ) ; note that because X is regular then ~ *

uniform cover of (X, Y). note that

, E < D},

(To show that

( CY~ 2 LJ {X - Y})*

refines

is again a

C~2" U{X - Y} is a uniform cover of X, (~2" ~ { x - Y}.)

For each uniform cover ~ of X, there exists a s i m p l i c i a l map

gc~ : J ( C ~ * ) which, on vertices E e ~ l * ' g~

satisfies

>K( L~ ) g~(E) e ~ l

and E < g~(E).

Of course,

is not determined by this condition but any two such simplicial maps have to

be contiguous and so, at the homology level, a unique homomorphism f~

= (g~),

:

H n ( J ( O * ) ) - - - ~ Hn(K( ~

))

is determined, which depends only on ~ and not on the p a r t i c u l a r choice of g ~ . Before going on, i t should be noted that the fact that g ~

is a s i m p l i c i a l map

arises from the fact that ~enever ~ is a f i n i t e subset of ~ I * ' and only i f the form

~{g~(E)

I Ee ~ }

# B.

~ is near i f

Also, since the set of a l l covers of

L~* is a cofinal subset of the set of a l l uniform covers of (X, Y), i t

follows that the fc~ form a homomorphism of the inverse spectrum.

For each uniform cover L~ of (X, Y), K ( ( ~ )

is a subcomplex of J( ~I ) so we

have the homomorphism k~ : Hn(K( ~ ))

>Hn(J( ~

))

induced by the inclusion map. Turning our attention now to the l i m i t groups, we have the projection homomorphisms u ~ : Hn(X, Y)

Hn(J( C~ ))

V~ : Hn(X, Y)

Hn(K((~I)),

and as well as the l i m i t homomorphisms f~: Hn(X, Y)

v X > Hn( , Y)

and k:

Hn(X, Y)

Hn(X, Y).

Consider the following diagram: Hn(X, Y)

Hn(J(~*))

f~

t~o~, > H n ( d ( ( ~ I ) )

kc~

----a.

Hn(K(651")) .

~c~

k~

k

> Hn(K( 01 ))

\

>

Hn(X, Y)

I t is clear that each of the inner triangles is commutative, because each homomorphism is induced either by a projection of refinements or by an inclusion map. To show that f o k = I, let x e vHn( X, Y) and compute as follows:

v

O)

f

k

x

=

=

x

f ~ k 6 ~ , VC~m x X

=

V ~ X.

Consequently, f k x = x. An equally pleasant computation shows that

k Ofoo = 1

and the proof

of the theorem is complete. For regular nearness spaces, the above theorem provides an a l t e r n a t i v e method v

of computing the Cech groups:

one can compute by means of the vascular theory.

I f X is a regular nearness space and Y is a dense nearness subspace of X and i f is a c o l l e c t i o n of subsets of Y which s a t i s f i e s is near in Y.

(3{clxA i A e ~

} ~ ~

then

This observation, together with the knowledge t h a t , in the above

s i t u a t i o n , the homology groups of X are the same as those of Y, indicates t h a t , instead of passing to the extension X and using the Cech theory, one could stay i n side Y and use the vascular theory. Of course, not every nearness space Y is a subspace of a topological nearness space X so, even i f X is the completion of Y, there may e x i s t c o l l e c t i o n s C~ of subsets of Y such that

dl {clxA I A e C21 } : ~ .

In such a case, i t might also be

advantageous to use the vascular theory. We w i l l now present an example using thevascular homology groups Hn(X, Y). Consider the Euclidean plane as a nearness subspace (= uniform subspace) of i t s Alexandrov one-point compactificaton.

Let X be the nearness subspace (= uniform

subspace) induced on the subset 1 1 { ( I , y)[ - I ~ y ~ I } l . ) { ( x , ~) [ 1 < x} t_J { ( x , - T) I 1 ~ x} , The completion of X is a c i r c l e S1 on a 2-sphere S2.

Thus, by the fact proved

in [ 2 ] , the homology of X is the same as that of SI . The point here though is that the homology of X can be computed without going outside X.

The d e t a i l s are as follows.

The set of a l l f i n i t e uniform covers of X is a cofinal subset of the set of a l l uniform covers of X. So, consider an a r b i t r a r y f i n i t e uniform cover ~ :~Y = {A e ~

of X.

Let

I A is unbounded}

and l e t ~ > 0 be such that {GC X I diam G < E }

refines ~ .

Let x+ be the supremum of the set { l } L W { x e R I f o r s ~ e y > 0 and f o r some A e 0 ) - ~ ,

(x, y ) e A}

and l e t x- be the supremum of the set {l}U{x Let ~

e R I for some y < 0 and f o r some A e ~ - ~ ,

be a set of i n t e r v a l s on X of diameter at most E such that

{(l,y)

I -I B

B)®¢(B, A)

where d O and d I are respectively induced by : ~BB®~(B ',A) and ~(C,B) ® eBB

~(c,B) ~ ~(B,B') ~ ¢(B',A)

~ ~(C,B') ~ ¢(B',A)

~(C,B) ® ~(B,B') ~ ~(B',A)

> ~(C,B) ~ ¢(~',A)

the maps ~BB' and and ~ .

~BB'

being respectively the actions of B(B, B')

on

This composition can be clearly extended to natural transformations of profunctors, and is associative and unitary up to isomorphism, if the unit associated to is the

•-functor

~(-, -) : h o P ® ~ ÷ ~V.

Right Kan extensions of profunctors. The main property of the (bi-)category of profunctors, is that there always exist right Kan extensions. If

¢ : ~ ---> ~

then the right Kan extension [[¢, X~ where -~

[X, Y]

[[¢, X]]

(D, B) =fA

is the value at

and of

×: & --~ X

along

~) are two profunctors, ¢

is given by the ends :

[¢(B, A), x(D, A)] Y

of the right adjoint of the functor

X : V ÷~f ~ that are the kernels of the maps to 7~ [¢(B, A), x(D, A) ] t, ) ~

to, t I : [¢(B, A ) ® &(A,A'), x(D,A') ]

25

where

tO

and

tI

are respectively induced (via the above adjunction) by :

[¢(B,A), x(D,A)] ® 0(B,A)®/A(A,A')

ev®)/

x(D,A)®A(A,A')

XAA' x(D,A')

[~(B,A'), x(D,A')] ® ~(B,A) ®/A(A,A') I®0AA, )[0(B'A')' x(D,A')]®0(B,A') eV~x(D,A') Thus

[[~, X]]

(D, B) is nothing but \V-Nat(0(B, -), x(D, -)).

Representability of profunctors. There is an embedding from 113, to

if- prof (/k, 8)

~r(A, B), the V-category of

the if-category of

associates to each functor

~f-functors between

•-profunctors between

F : A ÷ B, the profunctor

~

and

B

A

and

which

~ F : & ---+ 8 , where

O F(B, A) = ~3(B, F A). Definition. A profunctor tor

F : A+

B

such that

0 : A ---+ ~ is cal led representable if there is a func~ F

is naturally isomorphic to

0 •

Hence we have the classical result : Proposition. Let

F : ~ + ~3 and

G : IA + D

be two ~7-functors, then

point wise (i.e. preserved by all representable functors

G

admits a

D(D, -) : D ÷ ~f)

right

Kan extension, if and only if the right Kan extension of profunctors [[0 F, 0 G I]

is representable.

II. The ~-indexed limits. Let

11 be the ~¢-category with only one object

unit object of

~ . If

A

• , and such that

A ÷ 11 in general, so no canonical notion of limit. Let functor (i.e. aV-functor

11 (~, ~)

is a ~¢-category, there is no canonical

~÷~f)

and

F : A ÷B

a

is the

~V-functor

~ : /A ---4 11 be a

~¢-pro-

~f-functor. Then ([I] , [31) ,

we have the following. Definition. A projective ~-indexed cone from an object element of [[qb, 0 F ~

(B, ~) . The ~f-functor

~-indexed limit, if the right Kan extension object of

B

~,

(B, ~) ~-~3(B, ~-limF).

~ F~

(denoted by

F

B

of

B

to

F

is an

admits a projective (or inverse)

[[0, 0 F]] is representable by an

0-1im F) ; that is, if and only if Likewise a~f-functor

ductive or direct) ~-indexed colimit (denoted by

L : A°P+~

0-colim L), if

has an (inL °p : A ÷ 8 °p

has a 0-indexed limit. We shall call Examples. If

~ F

the indexation of the limit. is a V-functor : A-~ if , then it is clear that

since

[ X,V-Nat(~, F) ]~V-Nat(X ® ~, F) for each object

X

If

is a V-functor : /~P÷If , then ~-colim L = 0 ® L

where

L

functor

11 ---÷IA , since [~ ® L, X]

On the other hand, if

Y

= Nat(0, [L(-),X])

of

~-limF--if-Nat(0,F)

If. L

is seen as a pro-

for each object

is the Yoneda embedding : h °p ÷ V A

then

X of

V.

0-colim Y = 0

28

since

~(q,

L) = Nat(q, L)

by Yoneda ! , L

and Nat(q, ~ ( Y ( ) ,

is isomorphic to

Nat(Y(),

L)) = Nat(q, Nat(Y(), L)). Now,

L).

We shall study more precisely two particular important examples for our purpose. The Bousfield-Kan homotopy limits. Let

A be the well known category of ordered sets [n] = (0, I, 2, ..., n} , n • NN

and non decreasing maps. Let

S = set A°p

be the category of simplicial sets, which

is cartesian closed as is•very presheaf category. Let I be an ordinary small category, X

and

Y

two functors : I ÷ S , then Bousfield-Kan

[7]

define

Hom(X, Y)

as the kernel of : ]-[[X~,Y~] • | ] [Xi,Yi] iel ~ ~ i+i'eI J where the set

I(i, i')

. But

~ [Xi:.Yi,] = --[-[- [X i x l(i,i'), Yi' ] i-~i•I i,i'el J

is considered as a trivial (constant) simplicial set.

Therefore this Hom(X, Y) is nothing but vially enriched in

There is a canonical functor over

i

Nat(X, Y), where

I, X

and

Y

are tri-

S .

to each object

i

I/of

: I ÷ Cat

which associates the category

I . A category

C

I/i

being a particular simplicial set

(often noted

Ner C , but here we shall forget, as in

the functor

I/-

[7 ] , the notation Ner),

can be regarded as I ~ S .

Definition (Bousfield-Kan) : Let

X : I ÷S

be a functor, then

holim X = 14orn (I / -, X) . So, with the previous remark, holim X

is the

I/-

indexed limit of

X .

The lax limits of Gray. There are several ways to describe lax limits either from lax transformations [13] , [4 ]

which are generalized natural transformations, or as indexed limits

[ I 7] .

We shall give a new description of the indexation, which links the two points of view and is more appropriate to our aim. The Total category. Let

%

be the full subcategory of

i < 2 . Each functor

~

to

co < ]

A , formed by all the objects

[i]

÷ Cat determines a diagram : q0

~ t]

with : j.t 0 = j.t I = I,

cI

) C2 ql

q1"t0 = q0"tl '

n't0 = q0"t0 '

n'tl = q1"tl

such that

27 Let us define the total category Tot of this diagram

as the category whose objects

are the pairs (C, f : t0(C ) ÷ t I (C)) consisting of an object C of C0and a morphismf of

C1

such that j(f) = IC

and whose morphisms f'

and

q1(f) . q0(f) = n(f)

(C, f)~(C',

f')

are the morphisms

C g C'

such that

t0(g) = t 1(g) • f .

The total category has a clear universal property. Precisely let the inclusion Tot F

(In]

is the

Y2 : b2 ÷ Cat

can be seen as a category). Then if F: ~ + Cat , the category

Y2-indexed limit of

F, since clearly :

Tot F -~ Nat(Y2, F). The laxcones. Let

B

& ÷B

and

B

be 2-categories (i.e. categories enriched in Cat)land F a2-functor

. A lax cone from

an object

X

to

F

is the following data :

for each

A e ~

:

a morphism

X

for each

f : A+ A'

:

a 2-morphism

X

TA

> F(A) F(f) F

satisfying obvious coherences,

for

compositions

and

')

2-morphisms. These cohe-

rences are such" that a lax cone is exactly an object of the total category of the following diagram : ~B(X,FA) A with

t~ 'A,A'

[~(A,A'),B(X,FA') ] -- ~ ~ ~ [A(A,A') x~(A' ,A"),B(X,FA") ] ~--~-~,A,A' ,A"

t 0((TA)Ae A) = F(f) . TA ,

[qO(OA,A ')] A,A' ,A''(f'g) = F(g).eA, A,(f)

t I((~A)AeA ) = TA, ,

((n(@A,A,)~A,AtA,,(f,g)

= @A,A,,(~.f)

[J (@A,A ') ] A = @A,A(IA )'

[(qI(@A,A ')] A,A',A ''(f'g) = @At, A"(g)

Actually, this diagramm is determined by the right K~n extension of /~ / - : & - - - ÷ A

2 , where

l~/~

~ J_~ ~(A,~) ~(A,A')x~(A', A ~---A,A' wbere

d0CA f A' g ~3

=

A' g

d I(A f A' ~ ~) = A g÷f

~ F

along

is the internal category in Cat : ~ ~o ~)(** [ I A(A,A') x•(A' ,A") xA(A", ~) ~ A,A' ,A"

28

i(A ~f ~) =(A = A +f ~) P0(A+f A' ~ A" h ~) = (A f A' h~g c0 P I ( A I'+-A

aA"h-~) h ~)

m ( A f A' ~ A "

Whence t h e l a x l i m i t s We s h a l l d e n o t e

= (A'

~A" ~ ~)

A g-~f A" h ~)

=

are the

Y2 ® ~ / - l i m i t s .

Y2 ® A / -

by

Remark. T h i s new d e s c r i p t i o n

L(A) .

of lax limit leads to a generalization,

[6 ] , o f t h e B o u s f i e l d - K a n homotopy l i m i t s &/~

is an internal

simplicial

category in

: if

~

S and N e r ( A / ~ )

is a simplicial

studied in category,

is a cosimplicial

then

s p a c e . The

i n d e x a t i o n f o r t h e s e homotopy l i m i t s i s t h e p r o f u n c t o r

H(A) = A - - - * ! to generalize

, defined by the

H@A) (~) = D i a g ( N e r ( ~ / ~ ) )

replacement

scheme o f

[71

. This indexation allows us

to simplicial

categories.

The monads and t h e s e m i a d s . Let

~

be the 2-catego~

2-cells

~ : ~ + t ~.

with a single object

and

k t

~ : t2 ÷ t

=~.tk=t,

~.~

Then it is clear that a monad on a category such that its value at

~

limit of this 2-functor

is

C

Let

D

t

=~.

t~

is a 2-functor

C . It is well known too

:

from

[13] , [4]

~

to Cat

that the lax

is the category of algebras of the monad and that there

is a cotriple on this category of algebras, of

~ , g e n e r a t e d by a 1 - c e l l , and two

satisfying the well-known relations

that is an action of

D c°

(the dual

for the 2-cells). ~

be the sub 2-category of

semiad a 2-functor from

~

and a natural transformation Then the lax limit of are the pairs

a

D

generated by the 2-cell

to Cat, that is a category ~ : T2 ~ T

C

such that

(c, h • c ÷ T c )

such that

are the morphisms

The universal lax

cone

is

U(c, b) = c , and the 2-cell

b . ~(c) = b . T b

f: c ÷ c'

given by the

such that

forgetful

B : T . U + U

T

~ . ~ T = ~ . t ~.

semiad is the category of algebras

(c, b) ÷ (c', b')

~ . Let us call a

with an endo-functor

whose objects

and whose morphisms

f . b = b'

functor

given by

CT

. T f .

U : CT + C

6(c, b) = b : T c + c .

There is no longer an adjunction between C and C T, but it is clear too, that we have a functor

F : C ÷ CT

with

F(c) = (T(c), ~(c))

, such that

U . F = T . Fur-

ther more there is a natural transformation

q " F . U + I CT

Indeed

defines a natural transformation. Let

dl

q(c, b) = b : (T(c), ~(c)) + ( %

b)

be the 2-category with only one object

a 2-cell Proposition.

q : t-4~

~ , generated by a 1-morphism

. We can sum up this result in the following

There is a canonical action of

t

and

:

d7 on the category of algebras of a

29

semiad. III. The c~nonical action on G-indexed limits. These two last examples raise the question: is this fact general, is there always an action on the category of algebras! Let

~

be a

~g-category, A --~-~ ]I an

indexation. The profunctor ~ can be viewed as a functor Proposition. The

t-indexed limit of

nical action on each Proof.

~

A ÷ V •

is a ~-mono~d and this mono~d has a cano-

~-indexed limit,

~-lim t = U ~, ~]] = Nat(~, @) has an obvious structure of

F : ~ +~g t-lim ~

be a M-functor. So

~-lim F = ~ ,

acts on the ~-indexed limit of

~-functor and

L

M-mono~d. Let

and it is clear that

F . More generally, let

the t-indexed limit of

and the canonical action

F]] = Nat(t, F)

F :~ + B

be a

F . So we have ~(B,L) =Nat(t, ~(B,F -))

~-lim @ x ~(-, L) ÷ B(-, L)

which is equivalent by

the Yoneda le~raa to a morphism : t-lim ~ ÷ B(L, L) . It is easy to see that it is a morphism of V-monoids. Examples. The monad case. We have seen that the indexation limit of

y : ~op ÷ Cat~

tegory of the monad

k L~_~ I

is the lax

so that we can exhibit it as a monad on the Kleisli ca-

D(-, ~)

on

k(~, ~) . A simple but tedious computation of

its category of algebras shows us that this category is exactly mono~d structure is that of

~co(~, ~) and the

~co . The canonical action on the category of algebras

of a monad is the usual comonad. In the same way, we can study the semiad case. We must calculate the category of algebras of a semiad on the lax colimit of the semiad ~(-, ~) which is just action is

J/ (~, ~)

the

and the monoid structure is that of

on

~(~, ~) ,

J/ . The canonical

one described by the former proposition. More details will be

given in the proof of the next proposition. Remark. I choose this proof for sake of simplicity and quickness. But it is a very general result that (as in

the

case

of

V-functors [9] ) right Kan extensions

of profunctors are equipped with an action of the codensity monad (which always exists since we deal with profunctors) so that they can be factorized through the Kleisli category

[ ]6 ] of that codensity monad of profunctors, which in our case

has only one object and so is a mono~d. This general result is used in

[ 5 ] to

show in a very simple and categorical way that Kan extensions are shape invariant, so we could say that

~-lim t

is the "shape" mono~d of ~ .

IV. An appT,icatio~ to coherent homotopy. Following Dwyer-Kan gory

[10] , the standard resolution

C (a single object

t 2 = t)

•

F~ C

of the idempotent cate-

with a single non trivial morphism

is a simplicial category.

t

such that

30

But a category

is a particular simplicial set (via its nerve), so a 2-category is

a particular simplicial category. Now forthis category C , no composite of non identity maps is an identity, so F~ C see that this 2-category is just

is actually a 2-category and it is not hard to ~

.

We are now going to study the consequences of the higher homotopy coherences involved in the data of a simplicial functor from

B

to a simplicial category

which I keep on calling a semiad. In the special case

B

B = Top , I shall speak of

a coherent homotopy idempotent. The 2-pro-functor

L~)

simplicial profunctor. joint

K

: B ---+ |

indexing

lax limits can be considered as a

The simplicial embedding Cat L

preserving products, preserves

X

of

F

F : ~] ~ B

[14] ), being

the case of a 2-functor

ends and along

in a 2-category

B

F : [4 + IB

F

L(~), is still the 2-mono~d

we studied previously the action of

Y

F

has an L~H)-indexed limit L,

such that, if

c~(~)

a(~) . v = F(t) , and a 2-cell between

and the constant 2-functor on

iH(~, -)

then there

is the canonical projection v . ~(~) ~

given by

9(~) =~(t,-)

and

L(~) O

-)

~(t,-)j

:

~

"

~(~, -)

since we can verify that : @(t)

. @(~) I[-I(~, - )

= IH(U, - )

and

O ( t 2) = @(t)

. (9(t) ~ I ( t ,

. E(t, -)

- ) IH(!J, - )

= tI(p,

-)

= M(]~ . t 1~, - )

. I~(1~ t ,

-)

= tt(p

. ]] t ,

Whence a natural transformation : L(I0 ÷ H(~, -)

such that

~ T(~) = ~](t, -)

is the

between

: I](~,-) ÷~(~,-)

JH(~, -) = IH(ll,

IL .

on the li-

Y : B °p ÷ Cat~ . On the other hand, we have a lax cone

@(t)

In

a simplicial

L . Firstly let us consider the 2-enriched situation. We saw that

lax colimit of

~.

JJ. We have the B

Proof. The proof will be given by a careful study of the action of mit

. Thus

a simplicial functor.

v : F(~) ÷ L

L ÷ F(~) , we have

L(~)

L(l~)-indexed limits (homotopy limits in

L~H)-lim

Proposition. If the simplicial semiad exists a map

is also the

is the simplicial

following result about this action in the general situation of category and

L(~) -indexed

L(~)

considered as a simplicial functor. Therefore the sim-

plicial mono~d acting on the simplicial the sense of Gray

X

considered as a simplicial profunctor along

the lax limit of a 2-functor L(~)-indexed limit

has a simplicial ad-

exponentiations,

limits. So the right Kan extension of a 2-profunctor right Kan extension of

S

-)

31 if

• is the universal lax cone associated to the lax colimit of

there is a 2-1ax cone

6 between ~(~)O

and

Y . Furthermore,

T, given by

s i n c e t h e second members o f t h e f o l l o w i n g e q u a l i t i e s ~(~).T(N)o(t) are equal,

=

T(t).T(~)H(~,--)

,

b e c a u s e o f t h e c o h e r e n c e o f t h e l a x cone

~. Whence a 2 - n a t u r a l

transformation

L = Nat(LOt),F)

. The u n i v e r s a l

=

T(t).T(t)

S . Let L be t h e L (H) - i n d e x e d l i m i t o f F , t h a t L (N) - i n d e x e d cone o a s s o c i a t e d

We have a map o f s i m p l i c i a l Nat(~,F) :

and we v e r i f y

~(~).~

= F(t)

to the 2-cell

.

o(~) = Nat (T (~) , F) : Nat(L(tt) ,F) ---, N a t ( I ~ ( ~ , - ) ,F) = F(~) , and so on.

[q(t,-)

T with respect

d: T ( ~ ) a ~ I L ( H )

Now l e t us b e g i n w i t h a semiad F:14 ~ is

~(t).~(~)~(t,-)

s e t s w: F ( ~ ) - ~ L

F(e) = N a t ( I t ( ~ , - ) , F )

to o(t)

L

i s g i v e n by

= Nat(T(t),F)

, that is: , Nat(L(V),F) = L

, since

~(~) .~ = N a t ( T ( ~ ) , F ) , N a t ( ~ , F )

= Nat(T(~)~,F)

~ Nat(H(t,-),F)

= F(t).

F u r t h e r m o r e we have a 2 - c e l l b e t w e e n ~ . o ( ~ ) and 1 L g i v e n by N a t ( d , F ) and so F "splits"

at

L .

More g e n e r a l l y

l e t F : [ t * A be a semiad i n a s i m p l i c i a l

i n d e x e d l i m i t o f F and ¢ t h e u n i v e r s a l commutative d i a g r a m w i t h n a t u r a l

isomorphisms:

N a t ( T ( ~ ) , / A ( X , F - ) ) : N a t ( L 0 t )/A(X,F-)) A(X,o~) F u r t b e r m o r e we g e t a n a t u r a l A(X,F(~)) m

c a t e g o r y A. Let L be t h e L ( ~ ; -

L O t ) - i n d e x e d c o n e . Thus we have t h e f o l l o w i n g

* NatOH(~,-),IA(X,F-))

: /A(X,L)

, A(X,F(~))

f i n X) t r a n s f o r m a t i o n :

Nat(H(~,-)),/A(X,F-))

mat(~JA(X,F-))

Nat(L(H),N(X,F-))

~-~ N(X,L)

and so by t h e Yoneda len~na a morphism

v: F(~)--,I,

such that this natural

t i o n i s j u s t /A(-,v)

that o(~).v

is F(t), since

. Then i t

And now t h e 2 - ( n a t u r a l )

is clear

~.r(~)

cell

dJ atd

A(X,L) ~ Nat(L(i~) , A ( X , F - ) )

N a t ( r (~) ~,N(X,F-) )

A(X,L) ~ Nat(L(tl) ,N(X,F-})

transorma= ~(t,-).

32 determines, by Yoneda ! , a 2 - c e l l i n

N(L, L)

between

~ . ~(x)

and

1L •

Corollary. The homotopyidempotent ( i . e . idempotent i n the homotopy category H0-Top) a s s o c i a t e d to a coherent homotopy idempotent, s p l i t s . References. I. C. Auderset, Adjonctions et monades au niveau des 2-categories, Cahiers de Top. et G6om. Diff., XV (1974), 3-20. 2 J. B~nabou,

les distributeurs, Inst. Math. Pures et Appl. Univ. Louvain la Neuve

Rapport n ° 33 (1973). 3

F. Borceux and G.M.KelIy, A notion of limit for enriched categories, Bull. of

the AustralianMath. 4

Soc., 12 (1975) 45-72.

D. Bourn, Natural anadeses and catadeses, Cahiers de Top. et G~om. Diff. XIV

(1974) 371-480. 5

D. Bourn and J.M. Cordier, Distributeurs et th~orie de la forme, Cahiers de Top.

et G~om. Diff., XXI (1980) 161-189. 6

D. Bourn and J.M. Cordier, Une formulation g~n~rale des limites homotopiques,

Notes, Univ. Amiens (1980). 7 A.K. Bousfield and D.M. Kan, Homotopylimits, completions and localizations, Springer Lecture Notes in Math., 304 (1972). 8

J.M. Cordier, Sur la notion de diagran~e homotopiquement coherent , Proceedings

3~me colloque sur les categories Amiens 1980 (~ para~tre). 9

E.J. Dubuc, Kan extensions in enriched category, Springer Lecture Notes in

Math., 106 (1969). 10

W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, Journal of

P.A. Algebra, 17 (1980) 267-284 . 11

J. Dydak, A simple proof that pointed FANR-spaces are regular fundamental re-

tracts of ANR's, Bull. Acad. Polon. Sci. Math., 25(1977) 55-62. 12

P. Freyd and A. Heller

(in preparation).

13

J.W. Gray, Formal category theory, Springer Lecture Notes in Math., 391 (1974).

14

J.W. Gray, Closed categories, lax limits, homotopylimits, Journal of P.A. Alge-

bra, 19 (1980) 127-158.

15

F.W. Lawvere, Teoria d e l l e c a t e g o r i e sopra un topos di base, Mimeographed notes,

Perugia (1973).

Topology,13,(1974),

16

G.B. Segal, Categories and cohomology t h e o r i e s ,

17

R. S t r e e t , Limits indexed by category valued 2 - f u n c t o r s , Journal of P.A. Alge-

293-312.

bra, 8 (1976), 149-181. 18 M. Thiebaud, S e l f - d u a l s t r u c t u r e semantics and a l g e b r a i c c a t e g o r i e s , Dalhousie Univ., Halifax, N.S., (1971).

A REMARK ON C A R T E S I A N

H. B r a n d e n b u r g

I. A c a t e g o r y

A with

if for every A - o b j e c t For b a c k g r o u n d for c e r t a i n

and M.

finite p r o d u c t s

X the functor

concerning

aspects

cartesian

of a l g e b r a i c

c l o s e d n e s s and its

to

cited

there.

the c a t e g o r i e s

they c o n t a i n (for UNIF or UNIF,

spaces

[12]).

gorical

Aspects

tively

of T o p o l o g y

cartesian

respect

in TOP or UNIF.

(see r e m a r k

general

problem

category

of TOP or UNIF w h i c h

problem

is included

gorical

Topology

tain the

THEOREM

there

following

theorem,

point discrete space, As a c o n s e q u e n c e subcategory str o n g l y

of TOP m u s t

easily

and nega-

in the m o r e

reflective

closed.

recent

Note

survey

that

article

subthis

on Cate-

For the case of TOP we obbe proved

subcategory

in section

of TO__~P contains

I every c a r t e s i a n

consist

of c o n n e c t e d

space

of c a r d i n a l i t y

continuous

of X form an example contains

subspaces

here

car-

space w h i c h

can be a n s w e r e d

interested

w h i c h will

of T h e o r e m

ty is the only n o n - c o n s t a n t

TOP w h i c h

of usual

have

i.e.

2:

the two-

then it is not cartesian closed.

rigid H a u s d o r f f

all powers

TO_~P or UNIF

a non-trivial

11).

F.

on Cate-

subcategories,

is c a r t e s i a n

[7], P r o b l e m

If a reflective

I.

we are exists

in TOP are

fact,

Conference

whether

his q u e s t i o n

in H. H e r r l i c h ' s

(see

by this

a non-indiscrete

formation

Since

(c) below),

whether

Motivated

that

subcategories

that their p r o d u c t s

epireflective

to the

closed

1980 O t t a w a

containing

it is known

are c o r e f l e c t i v e

and Analysis)

closed

subcategories

are c l o s e d with products

(at the

importance and topolo-

and the l i t e r a t u r e

closed,

cartesian

products.

asked

[3].

TOP of t o p o l o g i c a l spaces and

are not c a r t e s i a n

from the usual

has r e c e n t l y

closed

[15],

All these c a t e g o r i e s

Schwarz

non-trivial

analysis,

[14],

and they have the d i s a d v a n t a g e

different

tesian

[5],

some nice n o n - t r i v i a l

see

cartesian closed

is called

topology,

algebra we refer Although

[2],

Hu~ek

X x - has a right adjoint

gical

UNIF of u n i f o r m

CLOSEDNESS

show that all c a t e g o r i e s

spaces. ~2

mapping

of a r e f l e c t i v e

only c o n n e c t e d

spaces

[4].

_AX o b t a i n e d

closed r e f l e c t i v e If X is a

(i.e.

the

identi-

from X into X) , then subcategory However,

~X of

one can

in this w a y are not

34

cartesian

closed.

cartesian

closed

We c o n j e c t u r e

rem shows

the v a l i d i t y

reflective

that

there exists

subcategory

of TOP.

of the c o r r e s p o n d i n g

no n o n - t r i v i a l

Our

second theo-

statement

for u n i f o r m

spaces. THEOREM

If a reflective subcategory

2.

indiscrete

space,

2. T h r o u g h o u t

this note all

and i s o m o r p h i s m - c l o s e d .

information

about

is i n t e r e s t i n g

PROPOSITION.

functor

I we will

Since A c o n t a i n s closed,

exists

T on the

a topology

(ii)

• is admissible,

Hence

where

Y is A - p r o p e r ,

i.e.

set C(Y,Z)

of c o n t i n u o u s

the e v a l u a t i o n

map

for every A - o b j e c t

is continuous,

topology

from

e:Y ×

(C(Y,Z),T)

~ Z

X and

for every contin-

w h e r e ~(f) (x) (y)=f(x,y).

exist

on C(Y,Z)

on C(Y,Z).

to A.

In order

X in A and a c o n t i n u o u s (C(Y,Z),~)

two spaces Y,Z

in A such

is not A-proper.

with

the usual

To this

topology

and

space Y × Yo' w h e r e Yo is the s u b s p a c e

e , both Y and Z b e l o n g

~

mappings

there

e(y,g)=g(y).

space of i r r a t i o n a l s

let Z be the p r o d u c t

~(f):X

that

of A - o b j e c t s

properties:

Y U {o} of the reals w i t h the usual metric.

topology

[8].

the countable

it is easy to v e r i f y

for each pair Y,Z

to show that there

admissible

end let Y be the

to

proposition

f:X × Y ~ Z the m a p p i n g

(C(Y,Z),T)

it suffices

that every

in TOP and contains

then

i.e.

is continuous,

~(f) :X ~

following

Then A is not cartesian closed.

the f o l l o w i n g

uous m a p p i n g

For addi-

we refer

of TOP which is closed with

a singleton,

if A is c a r t e s i a n

(i)

use the

Let A be a subcategory

respect to countable products

Y into Z w i t h

subcategories

to be full

C is reflec-

for itself.

infinite discrete space ~. Proof:

are a s s u m e d

A of a c a t e g o r y

has a left adjoint.

reflective

In the p r o o f of T h e o r e m which

subcategories

A subcategory

tive in ~ if the e m b e d d i n g tional

of UNIF contains a non-

then it is not c a r t e s i a n closed.

Being

homeomorphic

Let T be an a r b i t r a r y to p r o v e

mapping

that there

admissible

exists

a space

f:X × Y ~ Z such that

is not c o n t i n u o u s

we

start w i t h an a r b i t r a r y

to

35

continuous

mapping

g:Y ~ Z satisfying

for e a c h

y 6 Y, w h e r e

pl:Z

Consider

an a r b i t r a r y

point

hood.

By the

borhoods (yo,g)

continuity

V and V'

6 V'

cl V is not

tinct

points

X be the

Yo

~ Yo are

in Y w h i c h

of the

where

compact,

of Y

no c o m p a c t

map

that

that

exists

{Ynln

consisting

e w e can cl V c V'

a sequence

6 ~}

neighborfind

for

i=I,2}. of dis-

in Y.

non-negative

neigh-

and

(yn)n6~

is c l o s e d

of all

= 0

the p r o j e c t i o n s .

W = {z 6 Z I IPi(Z) I < I

there

in cl V such

subspace

has

evaluation

of Yo and a U 6 T s u c h

x U c e-l[W],

Since

Ip1(g(y)) I < I and p2(g(y))

~ Y and p2:Z

Now

let

numbers.

O

Being

homeomorphic

subspace retract

to Y, t h e

of X x y the of X × Y

continuous

(see

mapping

space

X belongs

space A =

({0}

[11],

II,

§26

f r o m A into

× Y)

has

a continuous

[

Z defined

that

an o p e n

neighborhood

there

exists

hand we conclude

from and

diction

shows

that

6 ~})

is a

Consequently

the

is c o n t i n u o u s .

an m 6 ~

that

Y=Yn

Then

there

~(f) [B] c U.

satisfying

e(Ym,~(f) (x))

is not

and

x y ~ Z.

e ( Y m , ~ ( f ) (x))

~(f)

2).

if x > 0

B of 0 in X such

=

(pl(g(ym)),mx)

f:X

(C(Y,Z),T)

an x 6 B and

(X x {Ynln

by

(pl(g(yn)),nx)

~(f) :X ~

As a c l o s e d

if x = O

extension

Suppose

U

Corollary

S g(Y)

(x,y)

to A.

mx

> I.

exists

Moreover On the o t h e r

= ~(f) (x) (y m) = f ( x , y m) 6 W that m x

continuous,

< I.

which

This

completes

contrathe

proof. Proof

of T h e o r e m

taining

the

countable

1:

L e t A be a r e f l e c t i v e

two-point

discrete

discrete

space

tue of t h e p r o p o s i t i o n .

space

e, t h e n A Hence

subcategory

{0,1}.

is not

we assume

of T O P

If A c o n t a i n s

cartesian that

closed

e does

con-

the by v i r -

not b e l o n g

to

A. Suppose is g i v e n

t h a t A is c a r t e s i a n by r e × r e : e x e

reflection

of e.

the ~ - r e f l e c t i o n and

that

of exe

Then

~ re × re, w h e r e

To v e r i f y

l_~In£er~ x {n}

closed.

this

always = re x

fact

has

the

I Jn6e{n}

one

the A - r e f l e c t i o n re:e only

form

~ re d e n o t e s has to n o t e

r(exe)

= re x r~

=

of e × e the Athat

l~In6ere x {n},

by t h e c a r t e s i a n

36

closedness Now

of A

consider

, where

the

I_~i d e n o t e s

continuous

f(n,m)

mapping

tinuous that

re-re[e]

fact,

n = m

and

m • 1

if

n • m

and

m = I

{O,1}

To prove Xo

% @ and

re = re[e]

(since

the

second

£ re-re[e]

such

h : r e ~ re d e f i n e d

~

observation {0,1}

[

re(1)

Applying

essentially

of r e x r e

we can show that

{x}×re

6 rear

if

g(x,x)

= 0

the

same argument

We conclude (a) T h e o r e m taining

- by

te space.

to our assumption. exists

if

n % I n

this

section

I implies

an mapping

that

In fact,

[9], T h e o r e m

1.1

rex{re(n)}

I

g(x,y)

g(x,x)

A cannot

every

it h a s

that

is i m p o s s i b l e .

subspaces

the argument

that

with

space X which

=

, which

to the

if

Repeating

Consequently

a finite

sian closed. which

[e].

in p a r t i c u l a r

a contradiction!

In

is a h o m e o m o r p h i s m

there

= I

of r e x r e w e o b t a i n

[e],

con-

We claim

Then the continuous

g(x,x)

I

spaces

that

= O.

if

g'x're'n~'~ ~ ,; = If 0

x,y

~ re

, contrary

assume

a unique = f"

every x 6 re-re[e].

h o r e = r e = i d r e o r e a n d h • idre

each x 6 rear

exists g o r xre

by f x

for

re:e

e 6 A

assertion

there that

= I for

that

that

that g(Xo,Xo)

h(x)

satisfies

such

g(x,x)

imply

i.e.

by

otherwise

that

would

6 A),

defined

if

g:rexre

in A.

~ {0,1}

I

to t h e p r e c e d i n g

mapping

f:e×e

= ~ I 0

According

the coproduct

again

for t h e

sub-

= 0 for e a c h p a i r

= 0 for

each x 6 rear

be cartesian

closed.

[e]-

[]

some remarks. reflective

is n o t

to c o n t a i n

- always

subcategory

indiscrete

cannot

the reflective

contains

of TOP

con-

be cartehull

the two-point

o f X, discre-

37

(b) If a r e f l e c t i v e s u b c a t e g o r y A of TOP contains a n o n - i n d i s c r e t e space X w h i c h is not TI, then it is not c a r t e s i a n closed.

In case

that X is not T this follows from the fact that A m u s t contain the o b i r e f l e c t i v e hull of X (e.g. see [10]) and hence a n o n - i n d i s c r e t e finite space.

If X is a To-Space,

spaces is c o n t a i n e d in A

then the c a t e g o r y of sober

([10], T h e o r e m

1.3).

In particular,

the

t w o - p o i n t d i s c r e t e space belongs to A. (c) No

non-trivial epireflective

sian closed,

s u b c a t e g o r y of TOP can be carte-

since it has to contain the t w o - p o i n t d i s c r e t e space.

This answers S c h w a r z ' s q u e s t i o n m e n t i o n e d However,

in the introduction.

a simpler proof of this fact results

from the o b s e r v a t i o n

that there exist z e r o - d i m e n s i o n a l T 1 - s p a c e s X,Y,Z and a c o e q u a l i z e r q:Y ~ Z such that i d x X q : X x y ~ XxZ is not a q u o t i e n t m a p p i n g in TOP (e.g. see

[I], Exa/nple 4.3.4).

E s s e n t i a l l y the same a r g ~ e n t

that there is no n o n - t r i v i a l e p i r e f l e c t i v e tegory of p s e u d o t o p o l o g i c a l

spaces

shows

s u b c a t e g o r y of the ca-

[13].

(d) Every e p i r e f l e c t i v e s u b c a t e g o r y of an e p i r e f l e c t i v e subcategory of TOP is r e f l e c t i v e in TOP. a/nple, to e p i r e f l e c t i v e

Hence T h e o r e m

I applies,

for ex-

s u b c a t e g o r i e s of the c a t e g o r i e s of Haus-

dorff spaces or c o m p l e t e l y regular spaces. 3. In order to prove T h e o r e m 2 let A be a r e f l e c t i v e s u b c a t e g o r y of UNIF c o n t a i n i n g a n o n - i n d i s c r e t e space X.

If A is c a r t e s i a n

closed,

l_~ldenotes the copro-

then X~xl In6 {n} = l_~In6 X~×{n}, w h e r e

duct in _A" jection,

For each n 6 ~ let in :X~x{n} ~ X~xi---in6~{n} be the in-

and let Pn:Xex{n}

Pn is u n i f o r m l y continuous, mapping

f:X~xi_~in6~{n}

~ X be defined by

((xi),n) ~ x n.

Since

there exists a u n i f o r m l y c o n t i n u o u s

~ X s a t i s f y i n g foi n = Pn for each n 6 ~.

M o r e o v e r there are two points x,y 6 X and a u n i f o r m cover U of X such that no element of U c o n t a i n s both x and y. c o n t i n u i t y of f there exist u n i f o r m covers

By the u n i f o r m

V of X and W of

{~in£~{n}

such that V×W refines f-1(U). It follows that V refines every -I Pn (U), c o n t r a d i c t i n g the fact that the subspace {x,y} e of X e is not t o p o l o g i c a l l y discrete.

Hence A cannot be c a r t e s i a n closed

w h i c h c o m p l e t e s the proof of T h e o r e m 2. It is w o r t h m e n t i o n i n g that our proof of T h e o r e m 2 m a k e s no use of star-refineraents of u n i f o r m covers,

hence T h e o r e m 2 is v a l i d

38

also for the category N E A R of nearness follows that no non-trivial reflective

subcategory

tion applies,

[6].

Moreover

subcategory

of NEAR is cartesian closed.

for example,

to the category

spaces

epireflective

This observa-

to the category of proximity

of contiguity

spaces

it

of an epispaces or

[6].

REFERENCES

Elements of Modern Topology, McGraw-Hill,

[I]

R. Brown, (1968).

[2]

E.J. Dubuc and H. Porta, Convenient categories of topological algebras and their duality theory, J. pure appl. Algebra I (1971)

New York

281-316.

[3]

S. Eilenberg and G.M. Kelly, Closed categories, in: Proc. of the Conference on Categorical Algebra, La Jolla 1965, ed. by S. Eilenberg et.al., Springer-Verlag, Berlin-New York (1966).

[4]

H. Herrlich, On the concept of reflections in general topology, in: C o n t r i b u t i o n s to E x t e n s i o n Theory of T o p o l o g i c a l Structures , (Proc. Sympos., Berlin, 1967), 105-114, Deutscher Verlag d. Wissensch., Berlin (1969).

[5]

H. Herrlich, Cartesian closed topological categories, C o l l o q u i u m Univ. Cape Town 9 (1974) 1-16.

[6]

H. Herrlich, A concept of nearness, (1974) 191-212.

[7]

H. Herrlich, Categorical Topology 1971-1981, in: General Topology and its Relations to M o d e r n A n a l y s i s and Algebra V (Proc. of the Fifth Prague T o p o l o g i c a l Symposium, Prague 1981), H e l d e r m a n n Verlag, Berlin, (to appear).

[8]

H. Herrlich and G. Strecker, Category Theory, H e l d e r m a n n Verlag, Berlin, (1979).

[9]

R.-E. Arch.

Hoffmann, der Math.

33

(1979)

[11]

K. Kuratowski, (1966).

[12]

M.D.

ed.,

n~chterner Rdume, Manus-;

Topology, Vol. I, Academic Press, New York,

Rice and G.J. Tashian, Cartesian closed coreflective subcategories of uniform spaces, (preprint). F. Schwarz, Cartesian closednes8, exponentiality, and final hulls in pseudotopological spaces, (preprint). U. Seip, Kompakt erzeugte Vektorrdume und Analysis, Springer Lecture Notes

[15]

4

258-262.

R.-E. Hoffmann, Charakterisierung cripta Math. 15 (1975) 185-191.

[14]

sec.

Appl.

Reflective hulls of finite topological spaces,

[10]

[13]

General Topol.

Math.

in Math.

273

N.E. Steenrod, A convenient M i c h i g a n Math. J. 14 (1967)

(1972).

category of topological spaces, 133-152.

CROSSED COMPLEXESAND NON-ABELIAN EXTENSIONS Ronald Brown School of Mathematics and Computer Science, University College of North Wales.

Philip J. Higgins Oepartment of Mathematics, Science Laboratories, Durham University.

and

Durham, U.K.

Bangor, Gwynedd, U.K.

Introduction

Crossed complexes may

be thought of as chain complexes with operators from a

group (or groupoid) but with non-abelian features in dimensions one and two.

We start

by surveying briefly their use. The definition of crossed complex is motivated by the standard example, the

otopy crossed conrplex ~ Here

~]~

of a filtered space

is the fundamental groupoid

hom-

~ : X 0 c X] c ... c X n c Xn+ ! c ... c X.

~I(X|, X0)

of

homotopy classes

rel i

of maps

(I, i) ÷ (X;, X O) , with the usual groupoid structure induced by composition of paths. For

n a 2 , ~n_X is the family of relative homotopy groups

p E X0 . map

For

n ~ 2 , there is an action of

~]~

on

~n(Xn, Xn_ ] , p)

6 : ~n~ + ~n_]~ ; there are also the initial and final maps

The rules which are satisfied by all such crossed complex (§]). complexes.

~

for all

~n~ , and there is a boundary 60

6 ] : ~|X + X 0

are taken as the defining rules for a

In particular, the rule

66 = 0

shows the analogy with chain

Of course the individual rules are connnonly used in homotopy theory, with-

out necessarily considering the total structure. By a

reduced crossed complex

C

we mean one in which

have been considered for some 35 years. [2].

CO

is a point.

These

They were called "group systems" by Blakers

He writes that he follows a suggestion of Eilenberg in using these group systems

to apply the homotopy addition lemma in his investigation of the relationship between the homology and homotopy groups of pairs.

His proofs involve a functor from reduced

crossed complexes to simplicial sets; the values of this functor have been shown recently by Ashley [I] to be

simplicial T-complexes, and Ashley has proved the hard theo-

rem that this functor gives an equivalence T-complexes.

N

between crossed complexes and simplicial

This equivalence generalises the well known equivalence of chain com-

plexes and simplicial abelian groups, due to Dold and Kan [27; Theorem 22.4], and the functor

N

generalises also the nerve of a groupoid, which we use in §3.

Reduced crossed complexes satisfying in each dimension a freeness condition were called "homotopy systems" by Whitehead [3], 32], and his main example was is the filtration of a CW-complex

K

by its skeletons.

~

where

The paper [3]] gives inter-

esting relations between homotopy systems and chain complexes with operators: we shall generalise these results to crossed complexes in [10]. the papers [30, 31, 32] is

reallsability.

In §]7 of [32] Whitehead sketches a proof

of a theorem announced in §7 of [3]], that if of finite dimensional homotopy systems, and

An overall consideration in

~ : C ÷ C' is a homotopy equivalence C

is realisable as

~

for some

40

CW-complex K + K' .

K , then

C'

is also realisable as

~'

and

~

is realisable by a map

The approach to simple homotopy theory in this section of [32] seems to have

Deen ignored and indeed its predecessor [31] is not widely read. Huebschmann, Holt and others (cf. [20, 17] and the historical note [26]) have shown how crossed complexes may be used to give an interpretation of all the cohomology groups

Hn(G; A)

of a group

G

with coefficients in a G-module

A .

Lue has explain-

ed in [24] how related ideas had been developed earlier for varieties of algebras, rather than just for groups.

However, the tie-up with classical cohomology was not

made explicit (cf. p.172 of [24]). We have given in [6, 7] a colimit theorem for the homotopy crossed complex of a union of filtered spaces.

This theorem includes the usual Seifert-van Kampen theorem

on the fundamental groupoid of a union of spaces; it also includes the Brouwer degree theorem

(~n Sn = ~ ) ,

the relative Hurewicz theorem, and a subtle theorem of J.H.C.

Whitehead on free crossed modules [31; §16].

The proof of the colimit theorem in [7]

involves in an essential way two other categories equivalent to crossed complexes, namely m-groupoids and cubical T-complexes [6, 8].

With simplicial T-complexes [l],

~-groupoids [9] and poly-T-complexes [22], there are now five categories known to be equivalent to crossed complexes, the proofs in each case being highly non-trivial. The papers [16, 18] give other work on crossed complexes. One of our aims here is to show how the homotopy addition lemma (which plays a key rSle in the work of Blakers [2] and of the authors [6, 7]) is also important in the cohomology of a group G . We do this by showing that the standard crossed resolution CG , which is constructed algebraically in [20] and applied further in [2]], in fact arises as

~BG , the homotopy crossed complex of the classifying space of

The boundary maps in

CG

G .

are determined by the homotopy addition lenmla.

Our further aim is an exposition of the Schreier theory of non-abelian extensions. Much

has been written on non-abelian extensions and cohomology,

(cf. [5, 12, 13

23] and the further references there), but it is notable that, while tnereare accounts in several books on group theory, texts on homological algebra remain largely silent on the subject, presumably because there is no known exposition using chain complexes, on which expositions of the abelian case are rightly based.

Here we show that the non-

abelian features of crossed complexes allow an exposition closer to the abelian case, involving morphisms and homotopies.

We strengthen the theory, by presenting an equiv-

alence of groupoids which on components induces the usual one-one correspondence of sets.

We also generalise the theory, to extensions of groupoids rather than just

groups, and to "free" equivalences of extensions. l.

Crossed Complexes We recall from [6] the definition of the category (here denoted

complexes. A crossed complex

C

(over a groupoid) is a sequence

XC) of crossed

41 60 "'" ---+ Cn satisfying (].I)

the following

C1

"'6> Cn-I

J> "'" ---÷ C2 ~

C! ~ 61

C0

axioms:

is a groupoid with

CO

as its set of vertices

and

60 , 6 !

as its initial

and final maps. We write

Cl(p, q)

for the group (1.2)

For

n e 2 , Cn

disconnected (1.3)

for the set of arrows

is a family of groups

groupoid

The groupoid

C1

over C 0) operates

(x,a)~-+ x a

Here if

Cn(p) ~ Cn(q)

if

We use additive

p

For

p

to

q

notation

n e 3

(p,q E C O )

and

C](p)

q

and

a £ C1(p,q)

Cn(P)(n

C! , where

, then

e 2)

for all their identity is a morphism

CI

Cn(P)

Cn(n e 2)

lie in the same component

for all groups

0

(equivalently, C n i s a

the groups

on the right of each

n ~ 2 , 6 : C n ÷ Cn_ l

the action of a =-a+x+a.

{Cn(P)}PEC0

and for

x E Cn(p) and

and we use the same symbol (l .4)

from

C](p,p). totally

are abelian.

by an action denoted

x a c Cn(q)

.

(Thus

of the groupoid

CI .)

and for the groupoid

CI ,

elements.

of groupoids

acts on the groups

C|(p)

over

CO

and preserves

by conjugation:

x (1.5)

= 0 : C n ÷ Cn_ 2

~

for

n _> 3 (and

~0~ = 616 : C2 ÷ CO

as follows

from

(1.4)). (1.6)

If C2

c e C 2 , then

as conjugation

6c

by x

In the case when We observe

operates

trivially

CO

= -c + x + c

~roupoid

as a

, or, simply, C](p)

n ~ 3

and operates

on

.

C

C2(p)

two as defining

over

for

(x, c e C2(P))

is a single point, we call

that the above laws make each

(Cl, CO)

Cn

c , that is 6c

we take the laws up to dimension

is a module

on

a

a crossed module

C2

over

CI(p)

;

crossed module over the

as a

crossed Cl-mOdule.

, and we take the laws

reduced crossed complex.

Let

n ~ 3 .

Then

(1.1) - (1.3) as defining

Cn(p)

Cn

as

module over the groupoid (Cl, c O ) , or, simply, as a Cl-module. A morphism f : C + D of crossed complexes is a family of morphisms of groupoids fn : Cn ÷ On , compatible of

C 1 , D!

on

with

Cn , D n .

the boundary

We denote by

maps

XC

Cn ÷ Cn-!

the resulting

' Dn ÷ Dn-l category

and the actions

of crossed

com-

plexes. By restriction (over groupoids). identity fl

of structure, Let

(as happens

Suppose union of the

erators

f : C ÷ D

throughout

are the identity,

we call

be a morphism

§5) we write f

Cl(P)

, p ~ CO .

[x] £ C 2 with

and of crossed

of crossed modules.

f

C , a set

Then we say

6[x] = hx

of modules,

as a pair

If

(f1' f2 ) "

f0 If

modules

is the f0

and

morphism of crossed Cl-mOdules.

a

given a crossed module

and for any other crossed for all

we have categories

.for all

Cl-module

C'

C

X

and function

is

the free crossed cl-module on gen-

h

from

x E X , if such elements and elements

x ~ X , there is a unique morphism

f : C ÷ C'

[x]' c C 2' of crossed

X

to the

[x] are given,

with

6[x]'

= hx

Cl-mOdules

such

42

that

f[x] = [x]'

for all

uced case [31]. [11]),

C2

is constructed,

x a E C2(q) +yb+x

for all

This d e f i n i t i o n becomes

to the group case, given

C1 , X

x ~ X , a ¢ Cl(60Xx ' q), and

q ~ C 0 , wita the usual relations

w h e r e these make sense, and witll C1

is

free on generators

free crossed C1-module on these generators be a crossed complex.

[15] of the groupoid

C1

A crossed complex C2

C

A crossed complex

C

If

C

is exact and

(or, equivalently,

G C

~i C

Px ' x E X.

is the quotient

subgroupoid

6C 2 .

The

(for some

(on some graph

% : X 2 + CI) , and for

XI),

n e 3 , Cn

Xn).

exact if for

is

is a groupoid,

n e 2 =

Im

then

C

(6

G .

Cn+ 1 --+ C n)

:

•

together with an isomorphism

with a quotient m o r p h i s m

crossed resolution of

called a

equal to the zero at

is a free groupoid

(6 : C n --+ Cn_l)

Ker

6 : C2 ÷ C I

n e 3 , the induced structure of #iC-module.

CI

is a free crossed Cl-mOdule (on some

6[x]

with

, x E X , if it is a

totally disconnected

C n , for

free if

is

C

[x] e C2(Px)

with

-x a

~ x a) = - a + % x + a

fundamental groupoid

Its

by the normal,

rules for a crossed complex give

is a free ~iC-module

Ix] = x 0

can be regarded as a crossed Cl-mOdule

Such a Cl-module

C

is given in

% , as t h e g r o u p o i d w i t h g e n e r a t o r s

A module over

Let

the usual one in the red-

(an exposition of w h i c h

and

a=yb-a+%x+a

trivial.

x ~ X .

Analogously

C1 + G

w h o s e kernel

is

Wl C ÷ G

6C2) is

free crossed resolution if also

It is a

C

is

free. Let

G

follows. groupoid w

be a groupoid.

Let on

X

generating

C2

÷ C I) is the G-resolution

w

is

G

~ : C1 ÷ G .

be a function to the union of the

closure of the image of Let

G

X , with quotient m o r p h i s m

: R ÷ C1

G .)

A free crossed resolution of

be any subgraph of

Ker ~ .

CI(p)

may be constructed

Let

R

(CI, CO)

be any set and let

(X; R, w)

determined by

w

.

is a Then

G-module of identities for the presentation (cf. [II]). + Cn ÷

... ÷ C 3 ÷ K

of

K

by G-modules;

: C 2 + C 1 to give a free crossed resolution

of

as

be the free

, p E C O , such that the normal

(The triple

be the free crossed Cl-mOdule

G

and let

G .

presentation of K = Ker(6

: C2

Choose any free

this may be spliced (Such a construction

into for groups

is used in [20, 21].) As explained

in the introduction,

homotopy crossed complex defined by the skeletons (This is due to Whitehead case follows. 71~ is

w~

7n X = 0

for

~ l ( X , x o) •

p

example of a crossed complex

space

~ .

X ; then

~

Let

, and the homology of Hn(Xp)

(cf. [32; Footnote n e 2) , then

w~

in dimension ~

(i.e.

is exact,

is the

b e the filtered space

from w h i c h

the more general

two is given in [II].)

Ker 6/Im 6) is for

, p E X 0 , where

41]).

~

is a free crossed complex.

[31; §16] in the reduced case,

A simple proof of freeness

~I(X, X0)

based at

key

of a CW-complex

phic to the family of groups X

a

of a filtered

Xp

In particular,

n e 2

is the universal if

X

Further isomor-

cover of

is aspherical

(i.e.

and so it is a free crossed r e s o l u t i o n

of

43

2.

The homotopy addition lemma This is a basic,

it expresses

but not so easy to prove,

the idea that "the boundary

Its formulation

involves

all the structural

and so for completeness Let and let n > I

An An

from

then determine vI

to

n-simplex

have its filtration

~I(A l, A~)

of the homotopy

with ordered

set of vertices

by skeletons

A rn "

vI

Then

o , say.

Intuitively,

of its faces".

crossed

complex,

is also written

a .

determines

~n(A n, A nn-l' Vn)

The face maps

3io ~ ~n_l ~n , and the map

respectively,

{Vo,Vl,...,Vn} , is for

The unique arrow of 3 i : A n-I + A n

u : A 1 ÷ A n , which

sends

v0 ,

uo ¢ Zl ~n .

(The homotopy addition le,~na) The elements

1.

Proposition

to

elements

Vn_ l , v n

theory.

is the composite

elements

cyclic group with generator v0

in homotopy

we state it here.

be the standard

an infinite

lemma

of a simplex

a may be chosen so that

the boundary : ~n(A n, A nn - l '

Vn) ___+ ~ n - l ( A ~ - I ' Ann-2' Vn)

is given by -81o + 800 + 320

if

n = 2 ,

300 - (830)u° - 31o + 820

if

n = 3 ,

n-I E (-l)lSi O + (-l)n(3no)U° i=O

if

n > 4

For a proof of this result, homotopy

3.

[29].

lemma is given for m-groupoids

A corresponding

cubical

form of the

as Lemma 7.1 of [6].

The standard crossed r e s o l u t i o n Let

of

addition

see for example

o

G

be a groupoid.

G , in which

of n-tuples

NnG

of elements

ui

The geometric

realisation

The simplicial

structure

homotopy write

crossed

CG

There

on

complex

2.

Let

G

of

G

X = INGI

~

for this crossed

Proposition

is a well-known

is the set of composable

NG

such that

simplicial

set

(ul,...,Un)

u i + ui+ !

is defined

is known as the classifying

induces

a structure

(for the skeletal complex and call

be a groupoid.

NG

elements

on

for

space BG

of CW-complex

filtration

[28], of

X)

on

the nerve

G n , i.e. 1N

i < n .

of

G .

X , and so the

is defined.

We

it the standard crossed resolution of

Then

is a free crossed resolution of

CG

G .

G

and has the following structure. (i)

CoG = G O ; CIG

is the free groupoid on the sub-graph

vertices and all the non-identity arrows of

G .

The basis element of

CIG

u E G*

ation is extended to

G

(ii)

C2G

corresponding to by setting

is written

(u,v) ¢ N2G*

consisting of all the [u] , and this not-

[0p] = 0p

is the free crossed CiG-module on generators ~[u,v]

for all

G*

: -[u + v| + [u] + Iv|

(the composab~e pairs of

G*)

.

[u,v]

E C2G(~Iv)

with

44

(iii) For

is the free G-module on generators

n >- 3 , CnG

for all (u l,...,un) ~ NnG* . We also let [Ul,...,u n] ~ CnG some

ui : 0 .

(iv)

6 : C3G ÷ C2G

be the identity at

~lun

[Ul, .... u n] E CnG(~lUn )

if

(Ul,...,Un) ~ NnG

and

i8 given by

6[u,v,w] = [v,w] - [u,v] [w] - [u + v,w] + [u,v + w] ,

for al~

(u,v,w) E N3G .

For

(v)

n >- 4 , 6 : CnG ÷ Cn_IG

is giVen by n-I i Z (-1) [Ul,...,u i + ui+i, .... u n] i=l

~[Ul, .... u n] = [u 2 ..... u n] +

+ (-l)n[ul, . . . ,Un_l ]

[u n]

.

D

This proposition follows from the homotopy addition lemma, the standard description of the face operators in if

G

NG , and the fact that

is a group, then Proposition 2 shows

homogeneous) crossed resolution of how

CG

G

CG

BG

as defined in §9 of [20].

6 : C3G ÷ C2G

We have now shown

should be noted; the values of this

are in a family of (generally) non-abelian groups. a crossed complex

A

Note that

arises geometrically.

The curious formula for

functor

is aspherical [281.

to be the same as the standard (in-

A(CG)

abelianises

C

a chain complex

AC

is the bar resolution of C2

There is a funetor assigning to

with operators from G

TIC

[I0]; for this

(cf. [25]), for the group case).

However,

and so loses information.

The 3-simplices of

NG

may be pictured as 3

u

+

0

(cf. p.12 of [25]). n > I .

Now

NG

v

~

u

I

is a T-complex in which every n-simplex is thin for

Every T-complex has a groupoid structure in dimension I, and the above picture

illustrates the 3-simplex used to prove associativity [I] of this groupoid. suggests the link between

4.

~ : C3G ÷ C2G

This

and associativity in extension theory.

Homotopies The notion of homotopy has a similar importance for crossed complexes to that for

chain complexes.

However, because of the more complicated structure of crossed com-

plexes, there are several possible conventions for the definition of homotopy, and there are also two levels of generality (corresponding to free and based homotopy in

45

the topological

case).

Our definition

lemma in the algebra of m-groupoids m-groupoids, Let

If If

(4.2)

be morphisms

and

Cop e Dl(fp,gp)

x c Cn(q)

: CI ÷ D2

of crossed complexes.

e n : C n ÷ Dn+l(n >- 0)

p ~ C O , then

n ~ 2 81

from the cubical homotopy addition

a topic w h i c h we hope to develop elsewhere.

f , g : C ÷ D

is a family of functions (4.1)

follows

[6], applied to a natural notion of homotopy for

, then

.

A ~omotopy

0

: f : g

with the following properties.

If

x e Cl(p,q)

, then

Ol x E D2(g q) .

0nX e Dn+l(g q) .

is a derivation

over

gl

, that is if

x + y

is defined

in

Cl

then O1(x + y) = (01x)gY + ely (4.3)

For

n e 2 , e n : C n ÷ Dn+ l

a ~ Cl(P, q) , x ~ Cn(p)

,

, y c Cn(q)

If

x e C1(p, q)

then

(4.5)

If

n ~ 2 , and

x E Cn(q)

gy = gl y .

where

ga = gla .

gx = -e0p + fx + 00q - (6elx)

(A similar definition, [23].

A homotopy p ~ CO

but with different

For further comments,

0 : f = g

conventions,

eq = e0q . is given in the reduced case

see Remark 4 at the end of the paper.)

which are used by H u e b s c h m a n n

if

00p

is an identity for all

(It is these homotopies, [20].)

For emphasis,

with different

the more general

called ~re~ homotopies.

e : f = g , e' : g = h

is defined by then

conventions,

f0 = go ) "

kinds of homotopy are sometimes If

where

is said to be tel C O

(so that in consequence

.

then

gx = (fx) eq - 8n_l~X - ~0nX ,

by Whitehead

gl , that is, if

, then

en(xa + y) = (enx)ga + 8ny (4.4)

where

is an operator m o r p h i s m over

are (free) homotopies,

their composite

~ = e + 8'

~0 p = Cop + 8~p , p ~ C O , and if n ~ I and x ~ C1(p, q) or x ~ C (q), e' n q . It is easily checked that # is a homotopy f = h .

~n x = e~x + (0nX)

In the next section we will be considering which are the identity on

CO = D O .

only crossed complex morphisms

C + D

Therefore we w r i t e

(C, D)f

and

for the groupoids w h i c h have such morphisms vely the free, and the

rel C O , homotopies.

are thus the respective

sets of homotopy

(C, D) as objects,

and w h o s e arrows

The sets of components

classes of morphisms

over

are respecti-

of these groupoids C O = D O , and they

are w r i t t e n respectively [C, D]f

5.

[C, D]

.

Non-abelian extensions Throughout

A

and

this section,

is totally disconnected

~t~nsion

of

A

by

G

G

(i.e.

and A

A

will be groupoids

is a family

is a pair A ~ E ~ G

A(p)

such that

GO = A 0

, p e A 0 , of groups).

and An

46

of morphisms

of groupoids

(5.2)

p

is a quotient m o r p h i s m of groupoids.

(5.3)

i

maps

p

A

i

p

are the identity on objects.

isomorphieally

onto

is a quotient m o r p h i s m means

for more details action of

E

see [15].)

on

A

~

A

A free equivalence

a crossed

is an isomorphism;

i

(large)

free equivalences

A .

~ E

if ~

Extf(G,

A) of

E

induces an

This can be extended

trivially

is (with the quotient m o r p h i s m

A

by

P

> G

G

is a commutative

diagram

G that

~

also is an isomorphism.

is the identity.

Here

Act A

is an isomorphism

on

A

and A

and the equivalences

Under our assumption Act A

of

this implies

the extensions

For any groupoid

q

in

E/Ker p ÷ G ;

Such

W e can thus form two

groupoids

both having objects

of

conjugation

which

i' ~ E' ~

equivalence

is an

an isomorphism

G .

A ~

induces

E-module.

of such extensions A

such that

p

: ... ÷ 0 ÷ 0 ÷ A ÷ E

a crossed resolution of

a free equivalence

Ker p . that

For such an extension,

making

to a crossed complex p)

and

the following properties.

E0 = GO

(That

and

satisfying

(5.1)

A

there is a groupoid

A(p) ÷ A(q) that

A

A)

,

G , but having arrows respectively

the

of extensions.

has the same objects

determine

Ext(g,

by

Act A

as

of actions on the vertex groups

A , and an arrow in

of groups.

Act A

There is a conjugation map

is totally disconnected,

from

p

to

~ : A ÷ Act A.

this map and the action of

a crossed complex ---+ 0 ---+ ... ---+ 0 ----+ A--~+~ Act A

w h i c h we w r i t e

XA .

olution

G , then the action of

~

of

(o, 1) : E ÷ X A isomorphism further

If

(where

A i

E

R> G

o : E ÷ Act A)

(~, ~) : X A ÷ X A

where

is an extension with associated E

on

.

A free equivalence

~ : Act A ÷ Act A

, Extf(G,

e : (CG, X A) The m o r p h i s m

e

since this result

details.

is given by

is the restriction

our m o r p h i s m

H, N, G

of

ef .

of standard

a $~ = ~((~-1a)8) ;

, .

W e give the proof only for

read our

for the group case, and

but with differences

G, A, E ; his factor set

k : C2G ÷ A ; his a u t o m o r p h i s m

a ~-+ a u

ef .

theory, w e do not give full

are given in [14],§|5.1

rather than free equivalence,

: for Hall's

A)

* Ext(G, A)

is a reformulation

[Some of the calculations

for equivalence follows

induces a m o r p h i s m

as in (*) induces an

There are canonical equivalences of groupoid8 ef : (CG, xA)f

Also,

by conjugation

o'n = ~o : E ÷ Act A .

T h e o r e m 3.

Proof.

A

crossed res-

of

N

for

in notation as (u,v) e N u ~ G

becomes

becomes

our

47

morphism

h : CIG ÷ Act A ; his choice

morphism

£ : CIG ÷ E ; his f u n c t i o n

A morphism

CG ÷ x A

+ Act A , k : C2G ÷ A

over

u ~-+ ~

GO = A0

such that

of coset r e p r e s e n t a t i v e s becomes our

~ : H ÷ N

k

becomes our d e r i v a t i o n

~ : CIG ÷ A . ]

is d e t e r m i n e d by a pair of m o r p h i s m s

is an operator m o r p h i s m over

h : CIG

h , and such that

the equations h~ = ~k , k~ = 0 hold.

(These equations are equivalent to the first two equations

of [14], and indeed

k~ = 0

in T h e o r e m 15.1.1

is, b y P r o p o s i t i o n 2, equivalent to the "factor set" con-

dition k[u + v,w] + k[u,v] h[w] = k [ u , v + w] + k[v,w] for all by

G

(u,v,w) ~ N3G*

.)

G i v e n such a m o r p h i s m

is defined by setting

set of pairs

(u,a)

E0 = GO

such that

and for

u E G(p,q)

CG ÷ X A , a n e x t e n s i o n

p, q e G O , letting

, a e A(q)

v e G(q,r)

, b ~ A(r)

.

The v e r i f i c a t i o n that

reader (cf. p.220 of [14]).

We write

Suppose now given two morphisms (h,k)

, (h',k')

as above.

B = 80 ' ~ = @I "

Then

Let

~

E

E(p,q)

of

A

be the

, with addition

(u,a) + (v,b) = (u + v, k[u,v] + a h[v] + b) for

,

E

,

is a groupoid is left to the

E = e(h,k) CG ÷ x A

over

G O , w h i c h w e w r i t e as pairs

@ : (h,k) = (h',k')

is a d e r i v a t i o n over

h'

b e a (free) homotopy, and if

and w r i t e

u e G(p,q) , v ~ G(q,r) ,

w e have h'[v] = -Bq + h[v] + Br - ~ [ v ] k'[u,v] = k[u,v] Br - a~[u,v]

,

.

A s t r a i g h t f o r w a r d c a l c u l a t i o n shows that k'[u,v] + ~ [ u , v ]

= -~[u + v] + k'[u,v] + (~[u]) h'[v] + ~[v]

(and this verifies that our d e f i n i t i o n of e q u i v a l e n c e agrees w i t h that on p.22! of [14]).

Define ef(8)

: e(h,k) ---+ e(h',k') (u,a) ~

Then by

ef(8)

(u, ~[u] + a Bq)

, u ~ G(p,q)

, a E A(q)

is an i s o m o r p h i s m of groupoids which, w i t h the a u t o m o r p h i s m

a ~-+ a Bq , a ~ A(q)

, defines a free equivalence of extensions.

. A ÷ A

Conversely,

given any

free e q u i v a l e n c e

arises in the above w a y if ÷ A

A

~ e(h,k)

> G

A

~ e(h',k')

> G

B : G O ÷ Act A

ined b y

n(u,0) = (u,~'u)

e(h,k)

complex

E

.

Let

h'

B(q) = ~]A(q)

, and : G ÷ A

~ : C|G

the f u n c t i o n

~'

def-

of

is equivalent to

.

Finally, w e show that any e x t e n s i o n some

is d e f i n e d by

is defined b y extending to a d e r i v a t i o n over

~ : CIG ÷ G

A _~i E -P+ G

A

by

G

b e the quotient m o r p h i s m and consider the crossed

obtained by trivial extension of the crossed E - m o d u l e

A .

Consider

48

the d i a g r a m C3G l I + 0

6 ~ C2G i Ik + ~ A

6 .> CIG i I~ + ~ E

A ~ A c t The crossed complex groupoid. (h,k)

CG

is free, while

So the identity on

is a m o r p h i s m

is defined by

G

CG ÷ X A

over

GO

A

is exact,

has a lift

G I I= % G

(~,k)

and both have

: CG -> E .

and an equivalence

G

Let

as fundamental

h = o~ .

of extensions

Then

e(h,k) -> E

(u,a) ~-+ ~[u] + ia .

Thus the crossed complex approach in non-abelian resolution

E

~

is successful

extension theory are so-to-speak

(a kind of universal

example)

because

compressed

some of the difficulties

into the standard crossed

and in particular

into the formula for

: C3G ÷ C2G • By standard homotopy Corollary

Let

4.

C

arguments,

we obtain from T h e o r e m 3;

be any free crossed resolution of the groupoid

G .

Then there

are equivalences of groupoids

Corollary

Let

Let

5.

e~ : (C, xA)f

> Extf(G, A),

e'

~ ext(G,

: (c, ×A)

N i--~ F -P-+ G

F-module

N .

G

Let

C

C ÷ ~

induces

n : F ÷ Act A

is injective. of groupoids

a set normally generating

Then a m o r p h i s m such that

h(r)

such that

C! = F

and

N = 6(C2)

5 is when P ÷ XA

A

. is centreless,

i.e. w h e n

is determined by a m o r p h i s m

is a conjugation

of

A

for each

r

in

N .

isation of Dedecker's work on non-abelian

"

.

also enable one to give a crossed complex version of a general-

be as above and suppose given a crossed ~0 = GO)

G

,

, (~, X A) --+ (C, X A)

special case of Corollary

The above methods

is free.

isomorphisms

(~, xA)f--+ (C, xA)f

: A ÷ Act A

A)

~ Ext(G, A)

be a free crossed resolution of

An interesting

F

obtained by trivial extension of the

~f(G,

e" : (~, xA)

Then the projection

D

Then there are equivalences of groupoids e~ : (~, xA)f ~

Proof.

.

be an extension of groupoids such that

P denote the crossed resolution of

crossed

A)

A H-~tension of

A

by

G

ether with a m o r p h i s m of crossed modules

cohomology ~-module

and extensions

A

is an extension

(where A-~

~ E ~

[12].

Let

G , A

is a groupoid with G

as above tog-

49

i

A

~E

I In fact if, by extending exes

~

and

of crossed

xHA

trivially,

respectively,

we regard

a function

then the above diagram

a co~ugation $

from

HO

xHA ÷ x~A

alence of

H-extensions

and a conjugation

in which

(~,~)

and

: x~A ÷ x~A

a groupoid

' q ~ ~0

(~,~)

such that

Ext,(G,

' such that Define a free equiV-

.

: ~ ÷ ~'

o'n = ~

generalisation

e is similar

on components

over the identity

•

: (CG, ×~A)

G , A , ~

~ ~(G,

A)

,

~ Ext'(G,

A)

.

(Dedecker's

result

is the bijection

results

that given a morphism

~ * xHA

to Oedecker's

A theory of extensions

internal

category

includes

the above equivalence

of groupoids,

to define similar

for the group case. ~ ÷ x~A

(where

e

If

X

is a CW-complex,

the aohomology of

ants of function

spaces

Remark 4. such that

A homotopy ft(Xn)

are discrete,

c Zn+ 1

for

complexes.

C

maps

fo' f|

We will

induces

do not include

and also extensions

are not used.

complex,

it seems reasonable

simply as

[~,

C] .

(A

to Postnikov

invari-

to have applications

invariant.

: ~ ~ ~

is a homotopy

prove elsewhere

a homotopy

the non-abelian

is a homotopy

theory,

in

cohomology.

n ~ 0 .

Consequently,

of

[19] he rel-

is given in [23], using

in E2] and applied

It would be interesting

of filtered

then such a homotopy

above for CW-complexes,

is a crossed

is developed

in [3].)

ft

C

complexes

X with coefficients in

theory of such a non-abelian

to Dedecker

Dedecker's

The results

and crossed and

On p.309 of is as in Coroll-

to be the coequaliser

of T-algebras

This generalises

of groupoids.

nor free equivalences,

~

2-cocycles.

and cohomology

for T-algebras.

idea for chain complexes

homotopy

crossed

objects

induced

are groups.)

has proved related

(for groups)

ates such morphisms

3.

G ,

Such free equivalences

ary 5), one can define an extension A ÷ E ÷ G by taking E __+ two ma~9 A--+ F m A (the semi direct-product). In a letter

Remark

on

of Theorem 3.

to that of Theorem 3. when

Huebschmann

[20] he shows

2.

for which there is

The (strict) equivalences are those

A) .

ef : (CG, x~A)f

Remark

(o,l) : ~ ÷ xHA

There are equivalences of groupoids

Theorem 6.

1.

compl-

is the identity.

We have the following

Remark

H(q)

(~,~)

~(a) = a Bq , a ~ A(q)

to be an isomorphism

(~,~)

form under composition

e

as crossed

is a morphism

to be an isomorphism

to the union of the

~(x) = -Bp + x + Bq , x e N(p,q)

by

these crossed modules

complexes.

Define

The proof

I

~fo = ~f|

cohomology

that if

X0

ft : X ÷ Z and

of morphisms

suggested

Z0

of

in Remark

3

50

R E F E R E N C E S O.

H. ANDO*, A note on the Eilenberg-MaeLane invariant, Tohoku

I.

N.K. A q~iLEY, Crossed oompl~e8 and T-~omplexe8, Ph.D. Thesis, University of Wales, (197fl).

Math. J. 9 (1957), 96-104.

2.

A.L. BLAKEKS, So~e relations between homology and homotopy groups, Ann. of Math., (49) 2 (1948), 428-46].

3.

R. ~ROWN, Cohomology with chains as coefficients, Proc. Lond. Math. Soc., (3) 14 (1964), 545-565.

4.

R. BROWN, On K~nneth suspensions, Proc. Camb. Phil. Soc., (1964), 60, 713-720.

5.

R. BROWN, Groupoids as coefficients, Proc. Loud. Math. Soc., (3) 25 (1072), 413-426.

6.

R. B~OWN and P.J. HIGGINS, The algebra of cubes, J. Pure Appl. A1 E. 21 (1981), 233-260.

7.

R. BI~OWN and P.J. HIGGINS, Colimit theorems for relative homotopy groups, J. Pure Appl. AlE. 22 ( 1 9 8 1 ) , 11-41.

8.

R. BROWN and P.J. HIGGINS~ The equivalence of w-groupoids and cubical T-complexes, Can. Top. G~om. uiff., (3e Coll. sur les cat4gories, d~di4 a Charles Ehresmann), 22 (1981), 349-370.

9.

R. BROWN and P.J. HIGGINS, The equivalence of crossed complexes and ~-groupoids, CaLl. Top. G~om. Diff. , (3e Coll. sur les categories, d4di~ a Charles Ehresmann), 22 ( 1981), 371-386.

10.

R. BROWN and P.J. HIGGINS, On the relation between crossed complexes and chain complexes with operators, (in preparation).

1|.

K, BROWN and J. HUEBSCHMANN, Identities among relations, in L~Dimenaiona~ and T.L. Thiekstun, Lond. Math. Soc. Lecture Note Series 48 (1982).

12.

P. OEOECKER, Les foncteurs Ext~ , H 2 4891-4894.

13.

P, OEDECKER and A. FREI, Gdn4ralisation de la suite exacte de cohomologie non ab~lienne, C.R. Acad. Sci. Paris, 263 (1966), 203-206.

14.

M. HALL, JR., Tile gheo~d of groups., MacMillan (]959).

]5.

P.J. HIGGINS, Ca~egorv~es c ~

16.

P.J. HIGGINS and J. TAYLOR, The fundamental groupoid and homotopy crossed complex of an orbit space, (these proceedings).

17.

O.F. HOLT, An interpretation of the cohomology group~

]8.

J. HOWIE, Pullback functors and crossed complexes, Cah. Top. G~om. Diff., 20 (1979), 281-295.

et

H2

TapoLogy, Ed. R. Brown

non ab~liens, C.R. Acad. Sci. Paris, 258 (1964),

~l~oupo~ds, van Nostrand Math. Studies, 32 (1971).

Un(G, M) , J. Alg., 60 (1979), 307-318.

19.

J. HUEflSCHMANN, Letter to P. Dedeeker, (4th June, 1977).

20.

J. NUEBSCHMANN, Crossed N-fold extensions of groups and cohomelogy, Comm. Math. Helv., 55 (1980), 302-314.

2].

J. HUEBSCtD4ANN, Automerphisms of group extensions and differentials in the Lyndon-Hoehsehild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334.

22.

D.W. JONES, Po~l-T-comp~exe8,

23.

R. LAVEND~OMME and' J.R. ROISIN, Cohomologie non-ab~lienne de structures alg4briques, J. Algebra, 67 ( 1 9 8 0 ) , 385-414.

24.

A.S-T. LUE, Cobomology o f groups r e l a t i v e

25.

S. MAGLANE, Topology and l o g i c as a s o u r c e o f a l g e b r a ,

26.

S. MACLANE, H i s t o r i c a l

27.

J.P. MAY, S~npl~cia~ objects in a~gebralc topologyj van Nos~rand Math. Studies II (1967).

Ph.D. Thesis, University of Wales, (in preparation).

to a v a r i e t y ,

n o t e , J . A l g e b r a , 60 ( 1 9 7 9 ) ,

J . A l g e b r a , 69 ( 1 9 8 ] ) ,

155-174.

B u l l . Amer. Math. S o c . , 82 ( 1 9 7 6 ) ,

1-40.

319-320.

28.

G. SEGAL, Classifying spaces and spectral sequences, Publ. Math. I.H.E.S., 34 (1968), 105-112.

29.

G.W. WHITEHEAD, Elemen~a of /~omo~op~ ~heoz,~, Graduate texts in Maths. No. 61, Springer, Berlin~eidelberg-New York, (1978).

30.

J.H.C. WHITEHEAD, Combinatorial hometopy I, Bull. Amer. Math. Soc., (55) 3 (1949), 213-245.

31.

J.H.C. W~ilTEHEAD, Combinatorial hometopy II, Bull. Amer. Math. Soc., 55 (1949), 453-496.

32.

J.R.C. WHITEHEAD, Simple homotopy type, Amer. J. Math., 72 (1950), 1-57.

* NO~.

Reference [0] continues work of [2J.

UN CRITERE DE REPRESENTABILITE PAR SECTIONS CONTINUES DE FAISCEAUX Yves DIERS D~partement de Math~matiques, U.E.R. des Sciences Universit~ de Valenciennes, 59326 VALENCIENNES O. Introduction. Etant donn~ un foncteur tions chaque objet

B

de

B

d'un faisceau ~ valeurs dans

~

et fibres dans

globales d~fini sur la catggorie dans

~. La cat~gorie

~

flexive dans la cat~gorie

U : $ ÷ ~, on d~termine dans quelles condi-

est isomorphe g l'objet des sections globales continues

~ais~A

~, universel pour le foncteur sections

des faisceaux g valeurs dans

~

et fibres

peut alors ~tre plong~e d'une fa~on pleinement fiddle cor~~ais~A

si bien que chaque objet de

~

peut s'identifier

son faisceau repr~sentant. On utilise la construction universelle des spectres, topologies sepctrales et faisceaux structuraux donn~e dans [6] et on est rameng ~ d~terminer dans quelle condition le morphisme canonique de chaque objet de

~

vers l'objet des sections globales con-

tinues de son faisceau structural, est un isomorphisme. On montre qu'une condition n~cessaire et suffisante est que le foncteur

U

soit cog~n~rateur finiment r~gulier.

Cette notion, plus forte que celle de foncteur cog~n~rateur prDpre [7] et plus faible qne celle de foncteur codense [12], est obtenue ~ partir des notions de famille monomorphique stricte on effective

E8] on r~guligre

cog~n~ratrice par monomorphismes stricts

~]

[5] de morphismes, de famille d'objets

et de morphismes de presentation finie

relative [7~, et est d~crite de plusieurs fa~ons diffgrentes. Dans certaines conditions, un foncteur est cog~n~rateur finiment r~gulier si et seulement si il est cog~n~rateur. Ainsi si et si

~

est une sous-cat~gorie de

~

B

est une cat~gorie arithm~tique

[5] et [15],

ferm~e pour les ultraproduits et dont les mor-

phismes sont exactement les monomorphismes de

~

dont le but est dans

$, alors

est une sous-cat~gorie cog~n~ratrice finiment r~guli~re si et seulement si une sous-cat~gorie cog~n~ratrice, d'un produit d'objets de

c'est-g-dire si tout objet de

~

~

est

est sous-objet

&. II s'en suit un th~orgme de representations par sections

continues de faisceaux qui contient t o u s l e s

thgor~mes de reprgsentations qui utili-

sent habituellement des versions g~n~ralis~es du th~or~me chinois sur les syst~mes de congruences. En appliquant les r~sultats ~ des foncteurs

U

oubli de structure ad~quats entre ca-

tegories d'ensembles munis de structures alg~briques, on obtient d'une part, de tr~s nombreux th~or~mes connus de representation par sections continues de faisceaux dont quelques uns sont d~taill~s ici, et d'autre part, des nouveaut~s parmi lesquelles la representation des anneaux commutatifs r~guliers formellement r~els par des faisceaux de corps ordonn~s, celle des groupes ab~liens sans torsion par des faisceaux de groupes abgliens totalement ordonn~s, celle des espaces veetoriels r~els par des faisceaux d'espaces vectoriels euclidiens ou par des faisceaux d'espaces vectoriels norm~s, celle des ensembles par des faisceaux d'ordinaux finis. Une originalit~ de ces derni~res

52

representations

est que les faisceaux repr~sentsntsont

ces topologiques

non "spectraux"

au sens de Hochster

en g~n~ral pour bases des espa[9] car non To-s~pargs

et ~ven-

tuellem~nt non quasi-compacts. On utilise les notations et les rgsultats de [5], ]. Foncteurs

cog~n~rateurs

de presentation IB

finie

~

tout morphisme

B/~

U-injective

des objets de de

B

si tout morphismes

core appel~es

U : /A ÷ ~. Un morphisme

au-dessous U

de

B. ll est dit

de

finie

[7]

U-injectif

si

se factorise ~ travers lui. Plus ggngrale-

de morphismes

g : B -~ UA

de

B

Les families monomorphiques

families monomorphiques

f : B -> C

s'il est un objet de presentation

~

vers

(fi : B -> Ci)ic I

l'un de ses membres.

dams

finie relative

g : B -~ UA

ment, une famille

On consid~re une cat~gorie localement

[7] et un foncteur

est dit de presentation

dams la catggorie

finiment rgguliers.

[6].

de m~me source de vers

U

IB, est dite

se factorise g travers

r~guli~res

de morphismes

strietes ou effectives

dans

de

~, en-

[81, sont ~tudi~es

[5].

|.0. D~finition. famille

Le foncteur

U-injective

U : ~A ÷ ~

de morphismes

est coggn~rateur

de prgsentation

finiment r~gulier si toute

finie relative de

~

est monomor-

phique r~guli~re. Rappelons

qu'un foncteur

au sens de Grothendieck tion

HO~B(f,UA)

U : /A ÷ ~

est dit coggn~rateur

[|4] si tout morphisme

: HO~B(C,UA)

-~ HO~B(B,UA)

est n~cessairement

isomorphique.

|.]. Proposition.

Si le foneteur

U

f : B ÷ C

propre de

~

[7] ou cog~n~rateur tel que l'applica-

soit bijeetive pour tout objet

est cog~ngrateur

A

de

/A,

il est cogg-

finiment rggulier,

ngrateur propre doric coggn~rateu_r. Preuve

: Soit

f : B ÷ C

un morphisme de

bijective pour tout objet finie,

la categoric

objets de

B/B

de presentation monomorphique

B/~

A

de

Soit

tel que l'application ~

l'est aussi et le morphisme finie au-dessus de lui

finie relative.

Chaque morphisme

rggulier.

Le morphisme

m,n : C ~ D

r~guliers.

deux morphismes

g : D + UA

f

f

est colimite filtrante des i.e. f = lim f. i-~+ i

Ii reste g montrer que mf = nf

gmf = gnf

donc

et soit gm = gn

f

k : D + K puisque

; il se factorise done ~ travers

k ; ce qui implique que

est bijective

et donc que

et par suite

k

est monomorphique

dams les representations

jamais d'adjoint ~ gauche, mais ils ont ngcessairement donn~ par les fibres des faisceaux repr~sentants. a un multiadjoint

g gauche et pour chaque objet

une famille universelle

de morphismes

de

avec

f. :B+C. l i

est ~pimorphique.

est bijective

Les foncteurs qui interviennent

soit

f. : B ÷ C. est U-injectif donc i I est alors monomorphique r~gulier comme colimi-

v~rifiant

vgrifie

HomB(f,UA)

~tant localement de presentation

de prgsentatiOn

te filtrante de monomorphismes

Tout morphisme

~

~. La cat~gorie

B

vers

leur COnoyau~

Hom~(f,UA) HO~B(k,UA)

m = n.

par faisceaux ne poss~dent

un multiadjoint

~ gauche

[4]

On suppose donc que le foncteur B

de U.

U

~, on note (~i :B÷UAi)icSpecu(~

53 1.2. Proposition. Si le foncteur

a un multiadjoint ~ gauche, il est cog~-

U :~ ÷ 8

n~rateur finiment r~gulier si et seulement s i i l de

~

existe une classe

de morphismes

telle que

(I) tout morphisme diagonalement universel de morphismes de

~

de source

B

B

vers

U

est colimite filtrante de

et

(2) toute famille U-injective de morphismes de

~ , est monomorphique r~guli~re.

Preuve : La condition n~cessaire est satisfaite en prenant pour phismes de presentation finie relative de 0

~

de morphismes de

universelle

B

0

la classe des mor-

~. R~ciproquement supposons qu'une classe

satisfasse (I) et (2). Pour chaque objet

B

de

~, la famille

(Ni : B ÷ UA.) est monomorphique. En effet, si T est un objet de pr~i ~ et m,n : T ~ B sont deux morphismes vgrifiant D.m = N.n i i i c Specu(B), alors d'apr~s (|), pour chaque i, il existe un morphisme

sentation finie de pour tout d. : B ÷ D. i i

de

~

au-dessus de

N. i

tel que

d.m = d.n. La famille l i

est U-injective, donc monomorphique r~guli~re d'apr~s (2). Par suite tat est aussi vrai pour un objet quelconque d'objets de presentation finie de

(Ni). Soit

morphisme

fk(i) : B + Ck(i)

d.1 : B ~ D.I

de

~

)

m = n. Le =~sul-

~, puisque celui-ci est colimite (Ni)

~tant monomorphiques,

route

est aussi monomorphique puisque plus fine une famille U-injective de morphismes

~. Chaque morphisme

~. se factorise ~ travers un i k(i) c K. D'apr~s (I), il existe un morphisme

avec

au-dessus de

(d i : B + Di)icSpecu(B)

~

(fk : B ÷ Ck)kc K

de presentation finie relative de

de

~. Les families

famille U-injective de morphismes de [5] qu'une famille

T

(di)icSpecu( B

N.,I qui se factorise ~ travers

fk(i)" La famille

ainsi obtenue est U-injective donc monomorphique r~guli~re.

elle est moins fine que la famille

(fk)kcK

et m~me r~guli~rement moins fine [5]

puisque route image directe de la famille Ii s'en suit que la famille Lorsque les objets de

A

(fk)kc K

(d.) est U-injective donc monomorphique. i est monomorphique r~guli~re (prop. 2.1 [5]).

sont des ensembles munis d'une structure alg~brique d~finis-

sable par une th~orie logique du premier ordre, la cat~gorie

$

est ~ ultraproduits.

Nous allons montrer que, dans ce cas, il suffit de consid~rer les families finies de morphismes. 1.3. Proposition. Si le foncteur ultraproduits d'objets de (I)

U

U :A ÷ ~

a un multiadjoint g gauche et rel~ve les

& ([6] 4.1), les assertions suivantes sont gquiValentes

:

est cog~n~rateur finiment r~gulier,

(2) toute famille finie U-injective de morphismes de (3) il existe une classe

D

~, est monomorphique r~guli~re,

de morphismes de presentation finie relative de

~

telle

que a) tout morphisme diagonalement universel de de morphismes de

0

B

vers

b) toute famille finie U-injective de morphismes de Preuve :

U

est colimite filtrante

et

(I) => (2):Soit

(fi : B ~ Ci)ic[],n]

est monomorphique rgguli~re

une famille finie U-injective de

54

morphismes de

~. La cat~gorie

(B/~) n

existe une petite cat~gorie filtrante

~tant localement de presentation finie, il ~

et un diagramme

d'objets de presentation finie de

(B/B) n

que les morphismes

sont de presentation

tout

fik : B + Cik

((fik:B + Cik)icE1,n~kcK

dont la colimite est

(fi). C'est-g-dire

finie relative et que pour

i £ El,n],

fi = ~ fik" Pour ehaque k ¢ ~, Is famille (fik:B ÷ Cik)ie[l,n] keK est U-injective done monomorphique rfiguli~re. Notons (fijk : Cik + Cijk' f~jk: Cjk ÷ Cij k)

la somme amalgam~e de

(fik : C + Cik, fjk : C + Cjk)

et

n

n

Pik :

R Cik + Cik la projection canonique. Le morphisme (fik) : B + ~ Cik i=I i=l est noyau des deux morphismes (f~jkPik)(i,j)e[l,n]2 et (f~jkPjk)(i,j)e[1,n]2 de f,. (f~j:C. " i ÷ C.., xj xj :C.j -~ n Cij) la somme amalgam~e de (fi : B ÷ Ci, f. : B -+ Cj) et Pi : ~ C. ÷ C. la J i=l i i projection d'indice i, alors par passage ~ la colimite filtrante suivant ~, le morN phisme (fi) : B ~i=IH C.x est noyau des deux morphismes (f~:pi)J (i,j)cE1,n]2 et source

n i~l Cik

(f'.'.p.) 2 lj j (i,j)e[l,n] la famille

(~Tn) (i,j)=(l,l)Cijk.

et de but

n ~ C. i=l i

de source

(fi)ieEl,n ]

est monomorphique

morphismes

de

r~guli~re. ~

la classe des morphismes de presentation

~.

(3) => (1) :avec la proposition de

(n,n) ~ C .. Cela implique que (i,j)=(],l) ij

et de but

(2) => (3) : est satisfait en prenant pour finie relative de

Si l'on note

~

].2, il suffit de montrer que toute famille U-injective

est monomorphique

r~guli~re.

Soit

(fk : B + Ck)kc K

une telle

famille. Supposons qu'il n'existe aucune sous-famille finie U-injective de Pour chaque partie finie que

K de K, notons D(K o) o se factorise g travers l'un des morphismes

Hi

D(Ko) # Specu(B) ,

D(~) = ~

et

D(KoO

(fk)kcK .

l'ensemble des i ¢ Specu(B) fk

avec

K I) = D(Ko ) 0 D(K l)

tel

k e Ko. Les relations montrent que les parties

compl~mentaires finies de

des parties D(K ) dans Specu(B) quand K parcourt les parties o ~ o K, forment une base de filtre sur Specu(B ). Soit F un ultrafiltre plus

fin. Ii existe un objet (UAi)icSpecu(B)

AF

suivant

de

$

tel que

torise ~ travers un morphisme

fk

avec

colimite filtrante et que le morphisme existe

I ¢ F

fk" L'inclusion

tel que le morphisme I C D({k})

r~sulte que la famille

k c K. Puisque fk

D({k})

de

NF : B + UA F

UA F = I ~

se fac-

icl~ UA.I est une

finie relative,

il

H UA i se factorise ~ travers iel D({k}) e F, ce qui est en contradiction dans

Specu(B)

appartient ~

poss~de une sous-famille finie U-injective

est monomorphique

r~guli~re. La famille

est done monomorphique

injectives. La sous-famille

d~fini

est de presentation

implique alors

(fk)kcK

(fk)keKo

(H i : B * UAi)icSpecu(B)

soit l'ultraproduit

(Ni)icl : B ÷

avec le fair que le compl~mentaire de

La sous-famille

UA F

F. Le morphisme canoniquement

(fk)kcK

F. Ii en (fk)kcK . o

de m~me que toutes les families U-

est r~guligrement plus fine que la famille o

55

(fk)keK

(2 [5])

puisque toute image directe de

(fk)k~K

est U-injective donc mono-

morphi~ue. De la proposition 2.1 [5], il r6sulte que la f~mille

(fk)keK

est monomor-

phique rgguli~re. ].4. Proposition. Si le foncteur

U

est codense []2], il est cog~ngrateur finiment

r~gulier. Preuve : Si &

U

est codense, tout objet

B

de

B

est limite de t o u s l e s objets de

au-dessous de lui, ce qui implique que la famille de t o u s l e s morphismes de

vers

U

B

est monomorphique r~guli~re (prop. 5.4 F5]). Toute famille U-injective de

morphismes de source

B

est plus fine que la famille pr6c6dente donc est monomorphi-

que ; elle 8st m~me rgguli~rement plus fine puisque toute image directe d'une famille U-injective est U-injective donc monomorphique (prop• 2.1

; elle est donc monomorphique r6guligre

~]).

1.5. Proposition.

Le foncteur

existe un foncteur

V : K ÷ ~

U :~ ÷ B

est cog6n~rateur finiment r~gulier s'il

tel que le foncteur

UV

soit cog~n~rateur finiment

r~gulier. Preuve : Toute famille U-injective de morphismes de presentation finie relative de m~me source de

~

est UV-injective donc monomorphique r~guli~re.

2. Le crit~re de repr6sentabilit~. 2.0. Th~or~me. Soit

U :~ ÷ ~

un foncteur tel que : I)

ment de presentation finie, 2) U diagonalement universel d'un objet

~

est une cat~gorie locale-

admet un multiadjoint ~ gauche, 3) tout morphisme B

pr6sentation finie relative de source

de B

~

vers

U

est colimite de morphismes de

diagonalement universels pour

U, 4)

U est

cog~n~rateur finiment r~gulier. Alors tout objet dans

~

B

de

et fibres dans

~

d~termine un faisceau

FB

Spe_cu(B)

~ valeurs

$, dont l'objet des sections globales est isomorphe ~

qui est universel pour le foncteur sections globales que le foncteur

de base

Bet

F : ~ais ~A ÷ ~ ; c'est-g-dire

F admet un adjoint g gauche pleinement fiddle. Si les conditions

I), 2), 3) sont satisfaites, la condition 4) est en fait n~cessaire et suffisante pour obtenir la conclusion. Preuve : Les conditions ]), 2), 3) sont les conditions d'applications du th6or~me 3.1 de E6] dont on utilise ici les notations et les r~sultats (cf. 3.0, 3.], 3.3, 3.4, 3.5). Soit

B

un object de

a) la famille universelle foncteur

U

est monomorphique puisque le

est coggn~rateur. Cela implique que les morphismes de

universels pour f, g : C ~ D

~. (H i : B + UAi)ieSpecu(B)

U

sont ~pimorphiques. En effet si

sont deux morphismes v~rifiant

on a ~3f6 = ~jg6

ce qui implique

(nj• : D ÷ UAj)jaSpecu(D)

~jf

=rljg

6 : B + C

~

est l'un d'eux et si

f6 = g6, alors pour tout donc

f = g

diagonalement

j e Specu(D),

car la famille

est monomorphique.

b) Montrons que le foncteur

D : ~(B) ÷ D(Specu(B)) °p

est une 6quivalence de cat6-

56

gories. II est surjectif sur les objets d'apr~s la construction de est fiddle puisque les morphismes de morphisme entre deux objets de ~(B) de

tels que

A'(B)

~(B). Soit

D(6) C D(6'). Notons

(6,6'). La relation

6tant 6pimorphiques, 6 : B ÷ C, 6' : B ~ C'

D(6|6) = D(6) N D(@') = D(6)

gonalement universel de presentation finie relative

@|

est U-injectif.

6

est plein.

(dk : (C,@) ÷ (Ck,@k))keK

D(@) = keK ~J D(6k)" La famille relative est U-injective.

6 : B ÷ C

PB(Specu(B))

dans

~(B), est un faisceau. On en d~duit

canoniques

B ÷ FB(Specu(B))

F

F : ~ais ~A ÷ ~

FB(D(6)) ÷ Fc(Specu(C))

F : ~ais ~

FB(Specu(B))

on d~duit

est pleinement

fiddle

est un isomorphisme.

que l'application

Specu(~ ) : Specu(C) ÷ Specu(B)

~.j : A.1 ÷ A.j

ouvert (prop. 3.3.6 [6]) la fibre de

l'isomorphisme

le morphisme canoniquement ~ ~. Alors

Soit

et

j. Le morphisme

Fc(Specu(@)) ~

A'(B)

est un faisceau sur

est monomorphique

au point

= B

et du

(U~j) ~i = ~.6. Puisj i

fibre de

est la fibre de F6

en

i

(F~)D(@)

r~guli~re.

Or on a

(6k : B ÷ Ck)ke K

1.2, il

de morphismes

ke~K D(6 k) = Specu(B) , doric, puisque (FB(Specu(B)) ÷

r~guli~re. Compte tenu des isomorphismes

e), la famille pr~c~dente est isomorphe g la famille

3. Un crit~re de reprgsentabilit~

FC

est doric

: FB(D(6)) ÷

~'(B). D'apr~s la proposition

Specu(B), la famille de morphismes

est monomorphique

:

est cog~n~rateur finiment r~gulier. Tout morphisme

est colimite filtrante de morphismes de

FB

(F6)D(6)

est un plongement hom~omorphique

suffit donc de montrer que toute famille U-injective de

d~fi-

j e Specu(C ). Posons

U~. : UA. ~ UA.. On en d~duit que le morphisme i i j est un isomorphisme. U

On

F~ : F B ÷ Fc(Specu(~)) ~

l'isomorphisme d6fini par

(F@)i : (FB)i ÷ (Fc(Specu ( 6 )))i

f) Montrons que le foncteur

@ : B ÷ C e &'(B),

est un isomorphisme.

Specu(B). Montrons que le morphisme

i = (Specu(6))(j)

FB(Specu(B))

et

I), 2), 3) que le foncteur adjoint ~ gauche

(Specu(6),F 6) : (Specu(B),F B) ÷ (Specu(C),F C)

FB(D(~k)))ke K

F B ~ PB

PB(D(6)) = C

~tant des isomorphismes,

FB(D(~)) ÷ Fc(Specu(C))

est un morphisme de faisceaux sur

Ni

Cela exprime pr~ci-

dgfini par

soit pleinement fiddle. Montrons d'abord que pour

ni par le foncteur adjoint ~ gauche ~

Fc(Specu(C))

telle que

~12])

le morphisme canoniquement d6fini

en

r~guligre.

PB : V(Specu(B)°P ÷ ~

e) Supposons maintenant avec les conditions

note

vers

de morphismes de presentation finie

Elle est donc monomorphique

que le foncteur adjoint ~ gauche au foncteur

au foncteur

~(B)

6'

= PB(D(IB)) = B.

d) Les morphismes

(Th. 1, p.88,

C'est donc

est un morphisme de

une famille de morphismes de

(dk : C ÷ Ck)ke K

s~ment que le pr~faisceau structural pour

D

la somme amalgam~e

implique que le morphisme dia-

6~1@~

dans la cat~gorie ~(B). Ainsi le foncteur

et il

deux objets de

(61 : C ÷ CI, 61 : C' ÷ C|)

un isomorphisme d'aprgs 4). Ce qui implique que

c) Soit

N(Specu(B))

il y a au plus un

(6k).

special pour les categories arithm~tiques.

Le th6or~me suivant contient les th~orgmes de repr~sentabilit~ de faisceaux qui utilisent habituellement

par sections continues

une version g~n~ralis6e du th~or~me chinois

57

sur les syst~mes de congruences. arithm~tiques, 3.0. Th~orgme. trice de

6

exactement Alors

A

So it

6

une cat~gorie arithm~tique

les monomorphismes

de

FB

de base

teur sections globales

: Soit

valences

R

on note

B

B

de

B

g

et

un objet de

vers

~

et

~. Notons

phismes diagonalement

B

de

L'ensemble ~

nalement universels pour

de

~

d~ter-

$, dont

F

admet un

R

~. Pour chaque

(DR : B + B/R)RcSpecu(B )

En effet, si

g : B ÷ X

la relation d'~quivalence de

h : B/R + X

B

est dans

est un m~rphis-

sur

tel que

B

~

~, donc

B/R

multir~flexive ~R : B + B/R

les morphismes

$. Ils sont aussi de presentation B

par un

de

o0

~. Notons R

est une

et o0

~R

est

est fermg pour les colimites finies dans

~tant monomorphiques,

de

g

est

II est imm~diat qu'une telle fac-

de la forme

A'(B)

engen-

h D R = g ; puis-

engendr~e par un nombre fini d'~Igments

universel

R e Specu(B) ,

vers

~

de

A'(B)

sont diago-

finie relative.

~tant colimite

Tout mor-

filtrante de morphismes

A'(B), la condition 3) du th~or~me 2.0 est satisfaite.

II reste g montrer que 6. Pour chaque objet phique puisque de morphismes

~ de

$ B

est une sous-cat~gorie de

¢og~nfiratrice dans ~

est monomorphique.

B, les morphismes

De la proposition

de morphismes

4. Applications. de representation foncteurs

Le th~or~me 2.0

de

A'(B)

7.11

[5], il r~sulte que

est monomorphique

r~guli~re.

1.3 en prenant comme morphismes

permet de retrouver de tr~s nombreux

oubli de structure ad~quats.

de classe

th~or~mes

II suffit de l'appliquer

connus g des

De nombreux exemples de foncteurs

I), 2), 3) du th~or~me

sont satisfaites

[6]. II reste au lecteur ~ d~terminer dans quels cas, le foncteur

rateur flniment r~gulier.

est monomor-

A'(B).

par sections continues de faisceaux.

U :$ + 6

pour lesquels les hypotheses

dans

de

finiment r~guli~re de

(NR : B + B/R)

~. Par suite toute famille U-injective

Le r~sultat d~coule alors de la proposition de source

coggn~ratrice

6, la famille universelle

toute famille finie U-injective

U

B

et fibres dans

que le foncteur

La famille

est donc une sous-cat~gorie

le morphisme

de

B

; on obtient ainsi une factorisation de

sur

de

et tout objet

l'ensemble des relations d'~qui-

(qR : B ÷ B/R)RcSpecu(B).

relation d'~quivalence quotient.

sont

~.

est dans

l'unique monomorphisme

l'ensemble des morphismes

B/~. Les morphismes

B/R

quotient.

$, on note

R e Specu(B)

de la famille

cog~n~ra-

&

et qui est universel pour le fonc-

Specu(B )

~, le morphisme

torisation est unique. ~ A'(B)

B

est universelle.

h : B/R ÷ X

~

est isomorphe g

le morphisme

est un objet de

morphisme

~

de

fiddle.

vers un objet de

dr~e par

de

g valeurs dans

dont l'objet quotient

NR : B + B/R

de morphismes

X

B

une sous-cat~gorie

dont le but est dans

Specu(B)

un objet de

sur

~

F : ~ais ~A + B ; c'est-~-dire

adjoint g gauche pleinement

que

6

et

et telle que les morphismes

est une sous-cat~gorie multir~flexive

l'objet des sections globales

me de

et El5].

ferm~e pour les ultraproduits

mine un faisceau

Preuve

Pour la d~finition et des exemples de categories

on peut se reporter ~ ~]

sont dorm's U

est cog~n~-

Nous en ~tudions quelques uns. Pour les categories

arithm~-

88

tiques (4.3 ~ 4.7), on utilise plutSt le th6or~me 3.0. avec lequel on est ramen6 montrer que le foncteur inclusion objet de

~

U :& ÷ ~

est cog6n6rateur, c'est-~-dire que tout

est sous-objet d'un produit d'objets de

&. Or c'est une propri6t6 souvent

bien connue dont la preuve repose essentiellement sur le lemme de Zorn. Les representations (4.8 g 4.12) sont nouvelles. Les faisceaux repr6sentants poss~dent l'originalit6 d'avoir pour bases des espaces topologiques non "spectraux" au sens de Hochster [9] car non To-s6par6s et 6ventuellement non quasi-compacts. 4.0. Repr6sentation d'un anneau cormnutatif par un faisceau d'anneaux locaux [2], [3]. Compte tenu de 7.0 [6], il suffit de montrer que le foncteur

U : Gocc ÷ S n e

est

cog6n6rateur finiment r6gulier. Un ultraproduit d'anneaux locaux 6tant un anneau local, le foncteur classe

~

U

relgve les ultraproduits. Utilisons la proposition 1.3 avec la

des morphismes de la forme

A ÷ A[a-l]. Soit

(A ÷ A~aTl])i~[l,n~ ~

mille U-injective. Pour chaque P c Specu(A), le morphisme -I A ÷ A[ai(p) ] avec i(P) c If,hi ; alors

vers un morphisme

A ÷ Apse

une fa-

factorise

~ tra-

ai(p) ~ P. L'id6al de

A

engendr6 par l'ensemble des 616ments id6al premier de ea famille

A, est ~gal g

(A ÷ A[a71])icD,n]a

a. pour i ~ [l,n] n'6tant eontenu dans aucun l A. La suite a I , ...,an engendre done le A-module A. est donc monomorphique r6guli~re (8.O~5]).

ainsi la repr6sentation classique de

A

par son faisceau structural

On obtient

A.

4.1. Repr6sentation d'un anneau commutatif par un faisceau d'anneaux ind6composables [3], El 3]. Soit

U : /Anclnd ÷/Anc

le foncteur inclusion (7.5 [6]). Pour un anneau

famille des anneaux quotients de l'anneau des idempotents de vers

U. Chaque morphisme

(A ÷ A/PA)

o3

P

A, est une famille universelle de morphismes de

A ÷ A/PA

A. Montrons que le foncteur

U

A ÷ A/Ae

o7

est coggn~rateur finiment rggulier. Un ultra-

ultraproduits. Utilisons la proposition 1.3 avec la elasse A ÷ A/Ae. Soit

est un idempo-

e

produits d'anneaux indgcomposables ~tant indgcomposable, le foncteur

forme

A

est colimite filtrante de morphismes diagonalement

universels de presentation finie relative de la forme tent de

A c ~nc, la

dgcrit l'ensemble des id~aux premiers

0

U

relgve les

des morphismes de la

P c Specu(A), il existe

(A ÷ A/Ae.). [',n] une famille finie U-injective. Pour chaque llXC I i(p) c [ ,3 tel que le morphisme A ÷ A/PA se factorise

travers

i.e tel que

A ÷ A/Aei(p)

tient ~ t o u s l e s

ei(p) c P. Par suite l'idempotent

id6aux premiers d'idempotents de

(A + A/Aei)ic[l,n ]

~ e appari=I i A ; il est donc nul. La famille

est done monomorphique r6guli~re (8.2 [5]).

4.2. Repr6sentation d'un treillis distributif par un faiseeau de treillis locaux Compte tenu de 7.9 [6], il suffit de montrer que le foncteur

U : ~rDLoc ÷ ~rD

~],~] est

cog~n~rateur finiment r~gulier. Ce foncteur rel~ve les ultraproduits puisqu'un ultraproduit de treillis locaux est un treillis local. Utilisons la proposition 1.3 avec la classe

~

principal de

des morphismes quotients de la forme E

engendr6 par

jective. Pour chaque

a. Soit

E ÷ E/(a)

(E ÷ E/(ai))iEEl,n~

~ c Specu(E) , le morphisme

E + E/~

o3

(a)

est le filtre

une famille finie U-infactorise g travers un

59

morphisme

E ÷ E/(ai(~))

a I V ... V an Donc

avec

i(~) e If,n], et par suite

appartient alors g t o u s l e s

ai(~) e ~. L'61~ment

filtres premiers de

E ; il est ~gal ~

I.

E/(a I) ~ ... ~ (an ) = E/(a I V ... V an ) = E. La famille (E ÷ E/(ai))ie[1,n] est

monomorphique rgguli~re d'apr~s 7.O.1, 8.5.1, 8.5.2 [5]. 4.3. Representation d'un anneau com~utatif r~gulier par un faisceau de corps commuta-

[3],

tifs

[10].

Le foncteur inclusion

U : ~c ÷ &ncReg (7.3 [6~) satisfait les hypotheses du th~or~me

3.0. En effet la cat~gorie gorie cog6n~ratrice de r6gulier

A ¢ ~ncReg

(A ÷ A/P)PeSpec(A )

SncReg

est arithm6tique (8.5 [5]), ~c

est une sous-cat6-

/AncReg puisque l'intersection des id~aux maximaux d'un anneau est r~duite ~ z6ro et donc la famille des anneaux quotients

est monomorphique,

~c

est ferm6e pour les ultraproduits et tout

sous-anneau r~gulier d'un corps cormnutatif est un corps. 4.4. Repr6sentation d'un anneau fortement r~gulier par un faisceau de corps [0_]. Le foncteur inclusion 3.0, la cat~gorie

U : ~ ÷ ~nForReg (7.4 [6]) satisfait les hypotheses du th~or~me

/AnForReg

6tant arithm~tique (8.5 [5]) et tout anneau A c /AnForReg

6tant un sous-anneau d'un produit de corps. 4.5. Representation d'un groupe ab~lien r~ticul~ par un faisceau de groupes abgliens totalement ordonn~s ~I]. Le foncteur inclusion

U : ~bTotOrd ÷ ~bRet (7.10 [6]) satisfait les hypotheses du

th~or~me 3.0, la cat~gorie

AbRet

~tant arithm~tique (8.5 ~]).

4.6. Representation d'un anneau commutatif fortement r~ticul~ par un faisceau d'anneaux totalement ordonn~s [II]. Le foncteur inclusion

U : IAncTotOrd ÷ ~ncForR~t

du th~or~me 3.0, la cat~gorie

(7.11

[4)

satisfait les hypotheses

AncForRet ~tant arithm~tique (8.5 [5]).

4.7. Representation d'un anneau commutatif rggulier fortement r~ticul~ par un faisceau de corps commutatifs ordonn~s Le foncteur inclusion

[11].

~cOrd ÷ SncRegForRet

(7.12 [6]) satisfait les hypotheses du

th~or~me 3.0. 4.8. Representation d'un anneau commutatif r6gulier formellement r~el par l'anneau des sections globales d'un faisceau de corps commutatifs ordonngs. On considgre la cat~gorie

~ncRegFormRl des anneaux commutatifs unitaires r~guliers

formellement r~els i.e. qui v~rifient l'axiome :

~Xl, ...,Xn,

1+x~ + ... + x n2

ble, et des homomorphismes d'anneaux et le foncteur oubli de structure AncRegFormRl. Le foncteur A £ SncRegFormRl (O) - I ~ P (x ~ P

ou

U

admet un multiadjoint ~ gauche. Le spectre de

relativement g

(1) P + P C P

vers

U

est l'ensemble des parties

(2) PP C P

(3) P O (-P) = A

P

de

o7

telles que

(A ÷ A/p N (=P))

P, est une famille universelle de morphismes de

U. La topologie spectrale est engendr~e par les parties

{P : -(a~ + ... + a~) e P}

A

(4) Vx C A, ~y c A(xy c - P=>

y c P)) (cf. 7.27 [6]). La famille des anneaux quotients

munis des ordres quotients de A

inversi-

U : ~cOrd ÷

a I ..... a n e A. Elle n'est pas

D(al,...,a n) =

To-s~par6e car toutes

60

les parties

P

qui d~finissent un ordre sur

(-P) = O, sont des points denses de

r~me 2.0 sont satisfaites. Montrons que id~aux maximaux

M

de

A

A i.e. qui v~rifient en plus (5) P

Specu(A). Les hypotheses (1), (2), (3) du th~oU

est coggn~rateur liniment r~gulier. Les

sont r~els, donc les corps quotients

A/M

rgels et par suite ordonnables. La famille des anneaux quotients de maximaux est donc une famille de morphismes de morphique, donc le foncteur duits. Chaque morphisme

U

A

vers

par ses id~aux

U. Or c'est une famille mono-

est cog~n~rateur. Le foncteur

A ~ A/p ~ (_p)

sont formellement A

U

rel~ve les ultrapro-

est colimite filtrante de morphismes quotients

A ÷ A/I

o~ I est un ideal de type fini de A. Si (A ÷ A/I ,...,A ÷ A/in) est une 1 famille finie U-injective, elle est monomorphique donc monomorphique r~guligre puisque

la cat~gorie foneteur

U

AncRegFormReel

est arithm~tique (prop. 7.O.1.

[~).

Ii s'ensuit que le

est cog~n~rateur finiment r~gulier.

4.9. Representation d'un groupe abglien sans torsion par le groupe des sections globales d'un faisceau de groupes ab~liens totalement ordonn~s. Le foncteur

U : &bTotOrd ÷ SbSTor (7.30 et 7.31 E ~ )

est cog~n~rateur finiment r~gu-

lier car il est surjectif sur les objets puisque tout groupe ab61ien sans torsion est totalement ordonnable. 4.10. Representation d'un espace vectoriel r~el par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels euclidiens. Montrons que le foncteur

U : ~ucl ÷ ~ec(~)

est coggn~rateur finiment r~gulier (7.32

[6]), en utilisant la proposition 1.2 avec la classe E ÷ E/X E/Xi)ic I

o~

X

e

Xi(l)+...+Xi(n)

et un suppl~mentaire

P la forme

X'

quadratique positive

q

q e Specu(E). II existe

sur

9. Soit

(E ÷

est de dimension finie done poss~de une base

avec E

E. Soit

i(1),...,i(n) c I. L'es-

de codimension finie. Tout ~Igment

x = Xlel+...+Xpep+X'

factorise


est un sous-espace vectoriel de dimension finie de

une famille U-injeetive de morphismes de

pace vectoriel

D

par

i(n+l) c I

Xl,...,~

c ~

et

x

de

E

el,...,

~tant de

x' c X', on d~finit la forme

q(x) x~+...+x 2. Alors ISO(q) = X', donc P tel que le morphisme quotient E + E/Xi(n+l )

E + E/ISO(q) , c'est-~-dire tel que

Xi(n+l)C

ISO(q)

donc tel que

(Xi(1)+...+Xi(n) N Xi(n+l) = {0}. II s'ensuit que la famille des morphismes quotients (E ~ E / X (• 1) , . . .,E ÷ E/Xi(n+l) ) prop. 6.0 [5], la famille

est monomorphique r~guli~re (8.4 [5~). D'apr~s la

(E ÷ E/Xi)ic I

est monomorphique r~guli~re.

4.11. Representation d'un espace vectoriel rgel par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels norm~s. Le foncteur

U : ~orm(~) ÷ ~ect(~)

(7.34 [6])

est eog~n~rateur finiment r~gulier

puisqu'il factorise le foncteur cog~n~rateur liniment r~gulier

Eucl ÷ ~ect(~) (prop.l~)

4.12. Representation d'un ensemble par l'ensemble des sections globales d'un faisceau d'ordinaux finis. Montrons que le foncteur

U : @rdfin ~ ~ns (7.36 ~])


lier en utilisant la proposition 1.2 avec la classe E ÷ E/R

o~

R

est une relation d'~quivalence sur

est coggn~rateur finiment r~gu-

E

engendr~e par un ensemble fini.

61

Soit

(E * E/Ri)ie I

Soit

R

une famille U-injective de morphismes de

la relation d'gquivalence sur

E engendr~e par

finiment engendrge. La relation d'~quivalence seulement si (x = y o u

%

sur

(lea classes d'~quivalences de

9. Soit i(I) ..... i(n)el.

Ri(1) U ... U Ri(n). Elle eat E x

d~finie par et de

y

x ~ y

suivant

si et R

sont

des singletons)) poss~de un ensemble fini de classes d'~quivalence. L'ensemble quotient E/~

peut ~tre muni d'une structure d'ordre total et eat donc en bijection avec un

ordinal fini. II existe alors

i(n+l) e I

factorise l'application quotient

tel que l'application quotient

E ~ E/b, c'est-~-dire tel que

deux relations d'~quivalence

R

lea relations d'~quivalences

Ri(1),...,Ri(n)

suit que la famille

et

~

Ri(n+l ) C

E + E/Ri(n+1) ~. Or lea

sont premieres entre-elles (8.3 [5]). Donc sont premieres avec

(E ÷ E/Ri(1),...,E ÷ E/Ri(n+l))

Ri(n+l). II s'en-

eat monomorphique rgguli~re

(8.3. [5]). D'apr~s Is proposition 6.0 de ~] is famille

(E ÷ E/Ri)iE I e s t

monomor-

phique r~guli~re. 4.13. Quelques contre-exemples. On montre facilement que lea foncteurs ~c ~ Snc, &ncDifLoc ÷ &ncDif, ~oc ÷ An

~om ~ ~nc,

E6~ ne sont pas cog~n~rateurs propres, donc

ne sont pas cog~n~rateurs finiment r~guliers et par suite ne donnent pas de th6or~mes de representations. REFERENCES [O] R.F. ARENS et J. KAPLANSKY. Topological representatio n of algebra_~s, Trans. Amer. Math. Soc. 63, pp. 457-481, |948. [I] A. BREZULEANU et R. DIACONESCU. Sur la duale de la cat~gorie des treillis, Rev. Roumaine. Math. Pures et Appl. 14, pp. 331-323, 1969. I~1 J.C. COLE. The bicategory of topo~ and Spectra, preprint. M. COSTE. Localisation, spectra and sheaf representation, Lecture Notes in Math. 753, Springer-Verlag. Berlin-New-York, 1979. ~] Y. DIERS. Familles universelles de morphismes, Ann. Soc. Sci. Bruxelles, 93, III, pp. 175-195, ;979. [5] Y. DIERS. Sur lea familles monomorphiques rgguli~res de morphismes, Cahiers de Top Geom Diff, XXI-4, pp. 44"I-425, 1980. [6] Y. DIERS. Une construction universelle des spectres, topologies spectrales et faisceaux structuraux, Archiv der Math, ~ paraltre. [7] P. GABRIEL et F. ULMER. Lokal pr~sentierbare Kategorien, Lecture Notes in Math. 221, Springer-Verlag, Berlin-New-York, 1971. [8] A. GROTHENDIECK, M. ARTIN, J.L. VERDIER. Th~orie des topos et cohomologie ~tale des schemas, Lecture Notes in Math 269, Springer-Verlag, Berlin Heideberg New-York, 1972. [9] M. HOCHSTER. Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, pp. 43-60, 1969. IOl P.T. JOHNSTONE. Rings, Fields, and spectra. J°ur" AIm" 49, PP" 238-260' 1977" ; K. KEIMEL. The representation of lattice-ordered groups and rings by sections in sheaves. Lecture Notes in Math. 248, Springer-Verlag, Berlin-New-York, 1971. 2] S. MACLANE. Categories for the working Mathematician, Springer-Verlag, New-YorkHeidelberg-Berlin, 1971. I I R.S. PIERCE. Modules over commutative regular rings. Mem. Amer. Math. Soc. 70, 1967. 14 H. SCHUBERT. categories, Springer-verlag, Berlin-Heidelberg-New-York, 1972. 5 A. WOLF. Sheaf representations of Arithmetical Algebras. Mem. Amer. Math. Soc. 148, pp. 87-93, 1974.

Kan

extensions

and

systems

Armin

Given

a diagram

problems: M:

P

When

K

P is M'

~ A? C a n

a similar

way

that

it

in

the paper

we

functors

systems

[F,K]~

fits

k A of

a right Kan

the

Frei

functors

extension

the

of

sole

in the d i s c u s s i o n

apply

general

the

theory

of

some

a given

The

first

for t r e a t i n g

naturally

consider

the

following

functor

M' ~ R a n K M be c l a s s i f i e d

inducing

imprimitivity? reason

we

RanKM

M satisfying

as the r e p r e s e n t a t i o n s

fied by Mackey's answered

M'

> T

of i m p r i m i t i v i t y .

one

of t h e

second.

to a s p e c i a l

are c l a s s i -

question

it a g a i n

in

has been

briefly A t the

situation

is

e n d of

in m o d u l e

theory. All

concepts

gory

and all

mula

as

in

u s e d are

V-concepts,

right Kan

where

extensions

V is a b i c o m p l e t e

are p o i n t w i s e ,

given

closed

cate-

b y the K a n

for-

(2).

In the d i a g r a m

(i)

p

K

P denotes of K,

> T

a small

given

by

} SK

E

category,

ISKI

D is the o b v i o u s E is the

D

=

K any

functor

ITI a n d b y S K ( X , Y )

extension

embedding,

m [p,v]OP

of t h e

identifying

a n d S K the = Nat

identiy

an o b j e c t

shape

category

(T(Y,K.),T(X,K.)).

on objects

to a f u n c t o r

X in S K w i t h

the

and

functor

T(X,K-). Let A be a c o m p l e t e RanKF

for all F in

of t h e

formal

called

indexed

FK = RanKF *)

category

Hom-functor limit

is g i v e n

Supported

[P,A])

by

(it a c t u a l l y a n d F: (see

P

[A],

suffices

) A a functor. or

) a n d the Y o n e d a

[B,K] lemma,

by

the F o n d s

National

that

Suisse

where

A contain

B y the d e f i n i t i o n that

the r i g h t

notion Kan

is

extension

83 (2)

FK(-)

FK admits

= HOmp(T(-,K),F).

a canonical F(~)

We r e c a l l be shape

extension

F K = FD w h e r e

= H O m p ( E ;;,F).

that a f u n c t o r w i t h d o m a i n invariant.

In turn,

F admits

T which

factors

a canonical

over D is said to

extension

F = FE

where A

which

F(*)

= Homp(,,F)

is c l e a r l y

continuous.

The o p e r a t i o n s and Y:

( )K,

(^) is just the P

> [P,V] °p.

RanyF(,)

(-) and functor

( ) extend Rany

faithful

= Homp(*,F)

one has that Y*(^)

embedding

for any c o n t i n u o u s

f u n c t o r M"' : [P,V] Op

~ M"' R a n y Y Z R a n y ( M "

(^)Y* ~

Id(Cont[[P,v]OP,A],where C o n t [ [ P , v ] ° P , A ] [[P,v]°P,A]

Y) as Y is codense;

consisting

= ~(*).

~ Id[P,A].

has M "

of

ways

Indeed

On the o t h e r hand,

category

in o b v i o u s

, w h e r e Y is the Y o n e d a

= HOmp([P,v]°P(*,Y),F)

As Y is f u l l y

to f u n c t o r s

of c o n t i n u o u s

~ A one

thus denotes

the

full sub-

functors.

A

We also o b s e r v e

= Homp((-,K)

that

((ED)*~) (-) = FED(-)

,F) = FK(-).

Summarizing

we have

Theorem

Let A be c o m p l e t e .

gramm

i.

= HOmp(ED(-),F)

Then,

with

the n o t a t i o n s

above,

the dia

64 A

( )

[P,A]

[P,A] commutes

up to n a t u r a l

Corollary M:

2.

> A if and o n l y

tinuous~

furthermore

functor

M = M"' Y is,

M K ~ M'

and M"' ~ ~.

Remark.

From

If a f u n c t o r M'(*)

that

isomorphisms.

M':

> A is of the

if it is of the

for a g l v e n

is,

the p r o o f

of the

M"i: [P,V] °p

exist

in

Hence

the

According candidates

a left

[P,V] °p. functors

When

theorem

Y))

~

adjoint, A :

1 the

M satisfying where

the

M'

unique

M'" R a n y Y of H O m p

and h e n c e

~ M K. As

some

M'"

con-

continuous,

one

the

satisfying

~ RanyM"' Y t h e n

we have

preserves

preserve

all

extensions

all

the

limits

that

is r e p r e s e n t a b l e .

limits.

M'" of M'

[P,V] °p is a r a t h e r

to c l a s s i f y

an i s o m o r p h i s m

[P,v]°P(A(A,~'Y), *)

~

M'"(*) ~ N a t ( M ''iY,*)

continuous

possible,

M'"

[P,V] (*,A(A,M'" Y))

V, then

~ M K for

we have:

definition

in C o n t [ [ P , v ] ° P , A ]

to T h e o r e m

it is p r e f e r a b l e ,

= M'" ED w i t h

> A satisfies

~ A(A,HOmp(*,M"'

M'" has

M'

form M'

form M' ~ M'" ED w i t h

up to i s o m o r p h i s m ,

~ Homp(,,M'" Y) . By the

A(A,M'"(*)

Cont[[P,v]°P,A]

y*

A functor

P

> Cont[[P,v]°P,A]

classify

large

candidates

the

category M by

func-

65

tors h a v i n g d o m a i n S

K

We call a f u n c t o r M": w i t h M'" c o n t i n u o u s [SK,A]

consisting

takes v a l u e s ponding

• We n e x t i n v e s t i g a t e

SK

) A a s y s t e m of i m p r i m i t i v i t y

and d e n o t e of s y s t e m s

in I m p s [ S K , A ] ;

corresponding

by I m p s [ S K , A ]

takes values

functor

The

we use the same symbol Imps[SK,A].

The

functor (

( ) clearly

restricted

and we use E*

~ Imps[SK,A].

of

for the c o r r e s -

f u n c t o r E*,

in I m p s [ S K , A ] ,

Cont[[P,v]°P,A]

if M" = M ' E

the full s u b c a t e g o r y

of i m p r i m i t i v i t y .

functor with codomain

to C o n t [ [ P , v ] ° P , A ]

that p o s s i b i l i t y .

With

for the

this n o t a -

t i o n we have:

Theorem

3.

In the s i t u a t i o n

of d i a g r a m

(1) a s s u m e

be small.

T h e n the f o l l o w i n g

statements

are e q u i v a l e n t :

(i)

E*:

Cont[[P,v]°P,v]

Imps[SK,V]

is an e q u i v a l e n c e .

(ii)

E*:

Cont[[P,v]°P,A]

Imps[SK,A]

is an e q u i v a l e n c e

complete

( ):

[P,V]

Imps[SK,V]


(iv)

( ):

[P,A]

Imps[SK,A]


(v)

E is codens e .

(±i) i m p l i e s

We have that

S K to

for all

A.

(iii)

Proof•

T, and h e n c e

(i) and

(E*o ( ? ) ) ( ~ )

(iv) i m p l i e s

= Homp(E~,?)

=

(iii)

for all c o m p l e t e

A.

trivially•

( 3 ) ( ~ ), that is, E*(^)

=

(-).

A

Since

( ) is an e q u i v a l e n c e ,

this e n t a i l s

(iii)

and

to

(v) :

(ii) is e q u i v a l e n t

that

(i) is e q u i v a l e n t

(iv). N e x t we show that

(iii)

to

implies

66

For 9:

any F,G:

P

[P,V](F,G)

> V we have

an i s o m o r p h i s m

) [P,V] (RanEE(F),G)

given

[P,V](F

G)

Imps[S K

V] (F,G)

Imps[S K

V] ( H o m p ( E ~ , F ) , H o m p ( E # , G ) ) ,

RanE([P

v]°P(G,E~))

via

(F),

(-), by the d e f i n i t i o n

by the K a n

[P,v]°P(G,RanEE(F))

by

of

(-),

formula,

as r e p r e s e n t a b l e s

preserve

RanEE

,

[p,v] ( R a n E E ( F ) , G ) , natural

in F and G, which,

isomorphism morphism,

~:

RanEE

hence

to show

the

of R a n E E

with

counit

I:

• Id.

that

we h a v e

jects

of I m p s [ S K , A ] , h e n c e

M'

ciated From

It is c l e a r

> A is of the is i m p r i m i t i v e

with

and,

from

again

and do c h o o s e

1 and

B(1)

is the u n i t

an iso-

that

is,

RanEE

= Id

for B = RanE(BE)

property as the

of RanE(BE) . This

second

arrow that

holds

is an i s o m o r -

E* hits

all ob-

it is an e q u i v a l e n c e .

f o r m M'

and

3 we

> Nat(AE,BE)

the d e f i n i t i o n s

that

= M"D with M"

M'.

Theorems

we m a y

Nat(AE,B(1))

if A is c o n t i n u o u s ,

so is E*.

that

that

of the u n i v e r s a l

phism

T

is ~ E ,

> A

E* ) N a t ( A E , B E )

isomorphism

If M':

of R a n E E

by a n a t u r a l

(ii) . If E is codense,

is an i s o m o r p h i s m ,

[P,V] °p

Nat(A,B)

afortiori

is i n d u c e d

Id E .... ~ E.

for any A:

is the

counit

(v) i m p l i e s

For B in C o n t [ [ P , v ] ° P , A ] and

The

lemma

E is codense.

It r e m a i n s counit

by the Y o n e d a

then

have

M"

is a s y s t e m

in I m p s [ S K , A ]

we


say asso-

67

Corollary M:

P

4.

A functor

) A if a n d o n l y

Corollary dense}

5.

The

in t h i s

M K ~ M'

The

M"

• A is of the

(-)


for a g i v e n there

imprimitive

is an M,

used comes

Taking

for K:

P

finite

groups

HCG

V = k-Mod,

the

to the

Mackey.

For

Theorem

3 has

from the

considered

functor as

( )

M' w i t h

some

if E is c o -

associated

the

see

kH

of s y s t e m s [F],

following

system

with

again

taken

as a o n e - o b j e c t

which

takes

t in T to t h e

P-modules

as

functors

(P-Mod) ° p a n d the

functor

o f S ° p to the P - m o d u l e (Z) = H o m p ( T , - ) .

in P - M o d ,

then

codense,

T. T h e

(-)

hence

S °p

between

sense

isoof

s ° P = ( E n d p T ) °p

ringhomomorphism

the u n i q u e

~ s°P-Mod

instance

generated By Theorem

T is d e n s e

ca-

the

[PV] ° p b e c o m e s

(-) : P - M o d


P ---mT

one-object

Interpreting

m (P-Mod) ° p t a k e s

(see for

L e t K:

ring

the

b y t.

the c a t e g o r y

(that is a f i n i t e l y

the P - m o d u l e

theory.

and D becomes

functor theorem

is t h e n

in the

of K is the

left multiplication

A Morita

if T is a p r o g e n e r a t o r

in m o d u l e

category

category

E:

taking

representations

Imps[SK,A]

as a f u n c t o r

f r o m P to Ab,

of

[M].

application

shape

and

to c o i n d u c t i n g Our

of g r o u p s :

group-algebras

categories,


[K] a n d

The

V = Ab.

-> k G of the

~ k-Mod.

tegories.

Here

kH

of r e p r e s e n t a t i o n s

as o n e - o b j e c t

interpreted

is t h e n

if a n d o n l y

u p to i s o m o r p h i s m ,

theory

corresponds

functors

category

details

K

be a r i n g h o m o m o r p h i s m ,

tor)

unique

~ T the embedding

considered

morphic

by

f o r m M' ~ M K for

a n d M ~ M".

terminology

of H,

T

if it is i m p r i m i t i v e .

functor

case


M':

[P])

is g i v e n says

projective 3 the

in P - M o d .

We

object

that genera-

functor thus

E

have

68

Theorem

6.

it is dense

Remark. ( ):

Let P be a ring and T a P-algebra.

If T is a p r o g e n e r a t o r

in P-Mod.

The d i s c u s s i o n

[P,A]

leading

~ Imps[SK,A]

to T h e o r e m

3 points


out that when

it g e n e r a l i z e s

a Morita

equivalence.

Bibliography

[A]

.C. Auderset, ries,

[B,K]

Adjonctions

Cahiers

F. B o r c e u x

[F]

A. Frei,

Shape

A. Frei

and induced

sion?, H. Kleisli,

[M]

G.W.

when

Mackey,

B. Pareigis, Teubner,

Soc.

2-categoXV,I(1974).

for e n r i c h e d

12

representations,

A question

L.N.

in Math.

Coshape-invariant

XXII-I

vol.

of limit

Math.

(1975).

to appear

in c a t e g o r i c a l

is a s h a p e - i n v a r i a n t

Springer

theorem,

in

functor

719

functors

shape

a Kan exten-

(1979). and M a c k e y ' s

induced

Cahiers

de Topol.

et G~om. Diff.,

representations

of groups

and q u a n t u m

(1981).

Induced

mechanics, [P]

A notion

Diff.,

des

Mathematicae.

representation Vol.

au niveau

et G~om.

Austral.

and H. Kleisli,

theory:

[K]

Kelly,

Bull.

Quaestiones [F,K]

de Topol.

and G.M.

categories,

et m o n a d e s

Benjamin-Boringhieri

Kategorien (1969).

und Funktoren,

(1968). Math.

Leitf~den,

SMOOTH STRUCTURES

by Alfred Fr~licher

A smooth

structure

on a set S consists

a set F c ]RS of f u n c t i o n s

of a set C c S

such that C and F d e t e r m i n e

tion t h a t FoC c C°°(~R, IR). The sets w i t h smooth ~if

we take as m o r p h i s m s

curves or,

equivalently,

with respect

The m a i n results the a t t e n t i o n

presented

to the e x c e l l e n t

show h o w c l a s s i c a l

calculus

finite dimensional[) singularities,

elementary

and are v a l i d

e.g.

cf

[3] or

properties

for each c a t e g o r y

For c e r t a i n m o n o i d s

[12].

explicitly

condition

Siciak

by d e s c r i b i n g

structure

result

(not~ecessarily

finds also o b j e c t s

with

in this direction. are those w h i c h we call the

is g e n e r a t e d

T h e y are easily o b t a i n e d

in a similar way, r e p l a c i n g B of B . For other examples,

the s m o o t h

structure

[7]. The p r o o f

yields

notion of C -maps.

between

the l i n e a r i t y the usual

by L a w v e r e ~ S c h a n u e l

for and

will be c o n s t r u c t e d

and for m a n y

w i t h the smooth the

theorems

of Boman

is f a c i l i t a t e d convex

of c a l c u l u s

functions

~-morphisms.

is a c a r a c t e r i z a t i o n

form

The

of smooth m a p s [2], of B o c h n a k

and

by u s i n g a m i n i m a l

spaces w h i c h does for

of the d i f f e r e n t i a l

results

and

of the function-spaces.

together

locally

A necessary

of this c o n d i t i o n

for any F r 4 c h e t - s p a c e

spaces w h i c h g e n e r a l i z e s

of c l - m a p s

closed.

first p r o v e d closedness

and that the C°°-maps are e x a c t l y

not r e q u i r e

nevertheless

and was

cartesian

the smooth curves

[I] and of H a i n

caracterization

is c a r t e s i a n

in [3]. The v e r i f i c a t i o n

in order to o b t a i n t h i s

Fr4chet

instance

: first we d r a w a n d then we will

manifolds

and cocompleteness.

which

In §3 it will be shown t h a t

between

there

the c a t e g o r y

was g~ven

The functor y i e l d i n g

C -Fr4chet-manifolds

basic

of~

within~one

(JR, IR) by any s u b m o n o i d

the m o n o i d C (JR, i~] is d i f f i c u l t

a smooth

to the

[4].

sufficient

Zame

set-up,

completeness

by any fixed set B and C

good w i t h respect

properties

but we shall not at all go here

ones,

yield a c a t e g o r y

here go in two d i r e c t i o n s

of C - d i f f e r e n t i a b l e

A m o n g the c a t e g o r i c a l

and

to the functions.

categorical

fits in this

of curves

each other by the condi-

structures

those m a p s w h i c h b e h a v e

~9

at a p o i n t but

and in p a r t i c u l a r

the usual

70

In r e c e n t y e a r s categories

several a u t h o r s u s e d w i t h a d v a n t a g e c a r t e s i a n c l o s e d

in o r d e r to d e v e l o p c a l c u l u s

for n o n - n o r m e d v e c t o r

c o n v e r g e n c e s t r u c t u r e s were u s e d in [6], c o m p a c t l y g e n e r a t e d

spaces

:

spaces in [ ~ ] ,

a r c - d e t e r m i n e d s p a c e s in [ii]. W h i c h c a r t e s i a n c l o s e d c a t e g o r y is the n a t u r a l one for c a l c u l u s ? If one w a n t s to study C~-maps,

the a n s w e r

vector spaces with a compatible

Some ideas and r e s u l t s in

s m o o t h structure.

seems clear

:

this d i r e c t i o n are g i v e n in the last section. The r e s u l t s understand~

p r e s e n t e d here are v e r y easy to f o r m u l a t e and to

h o w e v e r s e v e r a l p r o o f s r e q u i r e h a r d a n a l y s i s and thus can o n l y be

i n d i c a t e d in this e x p o s i t o r y article.

All p r o o f s w e r e c a r r i e d out in d e t a i l

in a s e m i n a r on s m o o t h f u n c t i o n s a t the U n i v e r s i t y of Geneva~ another

p r o o f of the t h e o r e m of Lawvere,

and the a u t h o r was p r e s e n t e d .


S c h a n u e l and Zame due to H. J o r i s

I w i s h to e x p r e s s m y g r a t i t u d e

for a v e r y

a c t i v e p a r t i c i p a t i o n and s u b s t a n t i a l c o n t r i b u t i o n s at this s e m i n a r in p a r t i c u l a r to G o n z a l o Arzabe,

Henri J o r i s and O s c a r P i n o - O r t i z .

§l T H E C A T E G O R Y ~ O F

S M O O T H S P A C E S A N D ITS E L E M E N T A R Y P R O P E R T I E S IR

A s m o o t h s t r u c t u r e on a set S is a c o u p l e F c I R S such that the

(C,F)

where C c S

"duality" C = D,F and F = D*C holds,

with

D.F = {c

: ]R ~ S~ foc 6 C°°(]R, ~R)

for all f £ F}~

D*C = {f

: S ~]R~

for all c 6 C}.

foc 6 C~( IR, JR)

A s m o o t h space is a t r i p l e

and

(S,C,F) w h e r e S is a set and

(C,F) is a

s m o o t h s t r u c t u r e on it. The morphisms

s m o o t h spaces form a c a t e g o r y ~ from

~.(C) c C' or,

(S,C,F)

so

(S',C',F')

for which,

are t h o s e m a p s d

by d e f i n i t i o n ,

: S ~ S' w h i c h s a t i s f y

e q u i v a l e n t l y ~*(F') c F.

The set of s m o o t h s t r u c t u r e s on a fixed set S is o r d e r e d manner

: (C,F)

m o r p h i s m from structure by C O

is c a l l e d finer than (S,C,F) to

(S,C',F').

(C',F')

and is o b t a i n e d as f o l l o w s (C,F)

IR

t h e r e is a f i n e s t

it is c a l l e d the s t r u c t u r e g e n e r a t e d

: F = D*C 0 ; C = D,F. S i m i l a r l y one has S : it is the c o a r s e s t

g e n e r a t e d by any set F 0 c ~ R

s t r u c t u r e w i t h F 0 c F and is o b t a i n e d as C = D , F 0 ; F = D*C. the s m o o t h s t r u c t u r e s of a fixed set S form a c o m p l e t e forgetful

in the u s u a l

if the i d e n t i t y m a p of S is a

For any set C O c S

(C,F) on S such that C O c C~

the s t r u c t u r e

the

f u n c t o r from ~ t o

It f o l l o w s t h a t

lattice,

that the

sets has a left and a r i g h t adjoint,

and that

71

~is

c o m p l e t e and cocomplete,

limits or colimits b e i n g o b t a i n e d

(as in the

c a t e g o r y of t o p o l o g i c a l spaces) by taking them in the c a t e g o r y of sets and then p u t t i n g the initial resp.

final structure on them• We note in p a r t i c u l a r

that the p r o d u c t of smooth spaces

(Si,Ci,Fi), i 6 I, is the object

(S,C,F}

w i t h S = × S. and C c o n s i s t i n g of those curves c : ~R ~ S w h o s e component i6I 1 c i : IR ~ S. b e l o n g 1 An o b j e c t

for all i 6 I, to C i.

(S,C,F) of ~ i s

called separated if for all a # b 6 S

there

exists f 6 F with f(a) ~ f(b).

~ denotes the r e s p e c t i v e full subcategory sep The inclusion functor has an obvious left adjoint and it follows

of ~ . that

~

is also c o m p l e t e and cocomplete. The forgetful functor from sop to sets still commutes w i t h limits, but not with all colimits. The one

sep point set w i t h its u n i q u e smooth structure is o b v i o u s l y

a final object of ~ ,

and it also yields a r e p r e s e n t a t i o n of the forgetful functor from ~ t o Another

important object is the triple

(~R, C~(]R, ~ )

, C°°(IR, ~R} ).

It will be d e n o t e d simply by IR and is generator and c o g e n e r a t o r of For any o b j e c t X = cx =

(SX, C x, F x) of ~ w e JcP(m,

x)

;

have

sets•

sep

:

~x : ~ C ~ ( x

,m)

.

The results of this section do not depend on the nature of the monoid C~(IR, IR) . As it was shown in [3], t h e y hold for the c a t e g o r y ~ g e n e r a t e d a n a l o g o u s l y by any m o n o i d M of maps of any set B to itself of BB). C a r t e s i a n c l o s e d n e s s of ~ h o w e v e r and sufficient condition was given.

(i.e. M a s u b m o n o i d

d e p e n d s on M; in [3] a n e c e s s a r y

In the following section we discuss this

c o n d i t i o n and its v e r i f i c a t i o n for the case B = R, M = C°°(IR, Z{) .

~2 CARTESI~ C~OSEON~SS OF dTPAND

~ep

F r o m the m e n t i o n e d p r o p e r t i e s of the o n e - p o i n t object it follows that if there is a functor

H :~ep

×~yielding

cartesian closedness,

it can be chosen such that the u n d e r l y i n g set of H(Y,Z) ~(Y,Z).

In p a r t i c u l a r one must get on ~ ( ~ ,

structure

(F,~)

such that, w i t h ~(x,y)

F = { ~ : ]R ~ C~°(]R, ~ )

; ~

JR)

then

is the function space

= C°°(]R, JR)

a smooth

: = 7(x) (y),

: m~IR

and } = D*F. Since t r i v i a l l y F c D,(D*F)

~]R a morphism }

the couple

(F,#) will be a smooth

structure iff D,(D*F) c F. Let us d i s c u s s the m e a n i n g of this condition. A c c o r d i n g to the d i s c r i p t i o n of p r o d u c t s i n ~ ,

a map G : 2

~ ] R is a m o r p h i s m

72

~

~

iff for all o,T 6 C°°( ~ , JR)

G is smooth along all smooth curves

one has Go(U,T) 6 C°°(~R, JR) , i.e. iff 2 . A c c o r d i n g to a remarkable

(O,T) of ~

t h e o r e m of Boman [2] this is equivalent to G 6 C ~ ( ~ 2, JR) yields a b i j e c t i o n F ~ C°° (

2, ~ )

and hence 7 ~

. From this and the d e f i n i t i o n of ~ as

D*F we get = {~ : C=°(]R, ~)---~IR; x ~ ~(G(x,-))

is in C ~ ( m ,

We call the elements of ~ smooth functionals explicitly;

for all G 6 C ° ° ( ~ m ) }

. One does not k n o w all of them

h o w e v e r for the linear ones one has

P r o p o s i t i o n 1.

The linear smooth functionals [9 have compact support

for each (p there exists a compact K fl I K

JR)

= f2 ~ K

( i.e.

of IR w i t h the p r o p e r t y

~ t0(f1) = [p(f2 )) and satisfy i9(lim f ) = lim ~ ( f ) n n n-~o n-~o

if fl,f2,..,

is a sequence in C°°(~R, JR)

such that for all k -> 0 the deriva-

tives

f(k)converge locally u n i f o r m l y for n ~ oo . This m e a n s that the linear n smooth functionals are exactly the d i s t r i b u t i o n s of compact support. This result is due to van Que and Reyes [13]

c o n s t r u c t i n g for c o n v e n i e n t subsequences f

, f nI

such that G(i/k,y)

P r o p o s i t i o n 2. I II

= fnk(Y) and G(0,y)

2 If G : IR ~

For all x 6JR,

G(x,-)

; it can be proved by

.... a function G 6 C~ (IR2 JR) n2

= lim fn(y)n-mo

satisfies 6 C~( ~ , ~ )

,

For all linear smooth functionals ~ • x ~ ~ G(x,-)

is in C°°(IR, IR)

then G £ C°°( m 2, JR) . This important result was proved by Lawvere,

Schanuel and Zame [12].

It can be p r o v e d by showing first the c o n t i n u i t y of G and its first o r d e r partial d e r i v a t i v e s ~1 G and D2G. This is quite d e l i c a t e for DIG and we found it useful to show first that DIG is p a r t i a l l y continuous in the second v a r i a b l e and ~IDIG is locally bounded. Once one has o b t a i n e d the continuous different i a b i l i t y of G, the p r o o f is completed by showing that DiG and ~2 G satisfy the same conditions I and II ; for D2G this is easy, and for DIG one m a k e s use of p r o p o s i t i o n i. F r o m p r o p o s i t i o n 2 is follows i m m e d i a t e l y that structure on C°°(~, ~ )

(F,~) is a smooth

. W e remark that for this it w o u l d be enough to prove

73

p r o p o s i t i o n 2 under the a s u m p t i o n that I holds and II holds for all smooth functionals q). However,

if one w o u l d a l l o w n o n - l i n e a r ones in the p r o o f that

G or ~l G or ~)2G are c o n t i n u o u s it w o u l d be

hard to get f u r t h e r , b e c a u s e one

does not haw~ the analogue of p r o p o s i t i o n 1 for n o n - l i n e a r smooth functionals

Theorem. for

The c a t e g o r y ~ o f

smooth spaces is cartesian closed. The same holds

sep The f u n c t i o n - s p a c e structure can be d e s c r i b e d e x p l i c i t l y

objects Y,Z of ~ o n e

d e f i n e s on the f u n c t i o n - s p a c e ~ ( Y , Z )

In fact, for

a structure

(C,F) by C = {d : ]R ~

~(Y,Z);

~

: ]RZ Y ~ Z a morphism}

F = D*C. Using that

(I',~) is a smooth structure on C°°(IR, JR) it is easy to show that

(C,F) is a c t u a l l y a smooth structure. D e n o t i n g the smooth space formed by the set ~ ( Y , Z )

with this structure by H(Y,Z)

it is s t r a i g h t f o r w a r d to show that

one has the universal p r o p e r t y : X -~ H(Y,Z) and

a morphism ~=~

: X Z Y ~ Z a morphism

this yields f u n c t o r i a l i t y of H and cartesian closedness of ~ . Since Z separated implies H(Y,Z) sep

separated, cartesian

closedness of

is o b t a i n e d by r e s t r i c t i o n of the functor H.

B e c a u s e in p r o p o s i t i o n 2 o n l y linear smooth functionals are used it follows easily that the structure of H(Y,Z) ~(Y,Z) support,

~[

of the form ~ ~ ~(f°~oc)

c 6 Cy =

9(IR,Y)

is g e n e r a t e d by the functions

w h e r e ~ is a d i s t r i b u t i o n of compact

and f 6 m Z = ~ ( Z

, m)

.

§3 THE S M O O T H STRUCTURE OF FRECHET SPACES AND MANIFOLDS. F o r ~ n the couple j OO~C(~, n )

, cOO( ~ n , ~ ) )

is a smooth structure. This

is not at all trivial, but it is equivalent to Bomans t h e o r e m [2], w h i c h says that a function o n ~ n

a l r e a d y quoted

is smooth if it is smooth along all

smooth curves. Using p a r t i t i o n s of u n i t y one gets a more general result for any finite d i m e n s i o n a l p a r a c o m p a c t C -manifold V, (c ~ ( ~ , V )

, C

:

(V, ~ ) )

is a smooth structure on V. So every such m a n i f o l d can be c o n s i d e r e d as a smooth space, and the C°°-maps b e t w e e n them are exactly the

~O-morphisms.

In order to get the same results for a greater class of v e c t o r spaces and m a n i f o l d s we need first of all a t h e o r e m w h i c h g e n e r a l i z e s B o m a n ' s result

;

74

this

theorem

and Hain

The useful

will

at t h e

same

time

aeneralize

results

of Bochnak-Siciak

[i]

[7]. following

(E,F w i l l

set-up

in t h i s

for c a l c u l u s

section

always

between denote

locally

convex

separated

spaces

locally

is

convex

spaces).

Definition. class

A map

f : E ~ F between

locally

convex

spaces

is c a l l e d

of

C 1 if for a l l x , h 6 E. df(x,h)

exists

: = w-lim i/~. I~0 a c o n t i n u o u s m a p df

and yields

By w-lim

we mean

Hence

the

df(x,h) 1 lim ~.

for all

1 [ F', We

Proposition

the

i.

limit

((lof) (x+lh)

-

: E × E ~ F.

in F w i t h

is c a r a c t e r i z e d

F' b e i n g

require


(unique)

(f(x+lh)-f(x))

to t h e w e a k

topology.

by

(lof)(x))

the topological

so l i t t l e

because

linearity

of d f ( x , - ) .

= l(df(x,h))

dual

o f F.

it is e n o u g h

to g e t the u s u a l

In f a c t o n e h a s

If f : E ~ F is of c l a s s lim l/l-(f(x+Xh] i~0

respect

- f(x))

C

1

properties,

:

, then

= d f ( x 0 , h 01

X~X 0 h~h 0 We

remark

(and n o t o n l y simultaneous strict

that

here

with

respect

limit

exists

differentiability.

Corollary.

to p r o v e

limit

is w i t h

to t h e w e a k shows

that

As e a s y

If f is o f c l a s s

In o r d e r useful

the

C

1

respect

topology!),

we a r e c l o s e

consequences

, then

proposition

to t h e t o p o l o g y

a n d the to w h a t

we have

f is c o n t i n u o u s 1 the

following

fact

that

of F this

is s o m e t i m e s

called

:

a n d df(x,-) "mean value

is linear, theorem"

is

:

Proposition

2.

interval

I c]R

(special

case

L e t A c E be c o n v e x into E related : d : c').

d(1) The proof For

by

application

f : EI×...xE n

(loc)'(~)

c,d

= l(d(~))

: I ~ E maps for all

for a n y ~ < ~ of I o n e h a s

6 A for ~ < ~ < ~ ~

is a s i m p l e a map

Then

and closed;

c(~)

- c(~)

6

~ 6 I and

1 6 E'

:

(~-~)-A .

of the H a h n - B a n a c h F the n o t i o n

of an o p e n

theorem.

"partially

of c l a s s

C I'' is

75

defined

in t h e u s u a l way;

i.e.

di~.=- Elx...XEnXE.l ~ F h a v e It f o l l o w s 1 C .

as u s u a l

that

the partial

to e x i s t

and

differentials

have

to be c o n t i n u o u s

f is o f c l a s s C 1 if a n d o n l y

in all v a r i a b l e s

if it is p a r t i a l l y

of

class

n now maps of class C : n+l f : E ~ F is o f c l a s s C if it is of c l a s s

Inductively Definition. c l a s s C n.

we d e f i n e

f is of c l a s s

For d n there operator

T behaves

C

1

a n d df is o f

C°~ if it is o f c l a s s C n for all n 6 N.

is a c h a i n

rule.

much better; Tf

It is c o m p l i c a t e d

it is d e f i n e d

and

for t h i s t h e

as

: E×E ~ FXF (x,h) ~ ( f ( x ) , df(x,h))

The

chain

rule then

says

a l s o gof, a n d T n ( g o f ) Since

: if f : E ~ F a n d a

: F ~ G are of class C

n

, then

= TncroTnf.

for f of c l a s s

C

1

t h e m a p df is l i n e a r

(and c o n t i n u o u s )

in t h e

s e c o n d v a r i a b l e , o n l y t h e f i r s t p a r t i a l d i f f e r e n t i a l is of i n t e r e s t a n d y i e l d s 2 2 2 a m a p D f : E x E X E ~ F. D f e x i s t s a n d is c o n t i n u o u s iff f is of c l a s s C , a n d t h e n D 2 f (x,_,_)

is b i l i n e a r

a n d D n+l as t h e

first partial

class

C n if a n d o n l y

Dnf(x,-,...,-)

if D l f ..... D n f

to s h o w t h a t

existence

D n-1 f. If we

suppose

symmetric.

differential

is n - m u l t i l i n e a r

If we w a n t must verify

and

and

a map

continuity

'

In ) ~

on d e f i n e s

D 1 f as df

of Dnf.

shows that

f is o f

One

a n d are c o n t i n u o u s ,

f of c l a s s C n-1 of the g

admits

first partial

basis

L e t E be m e t r i z a b l e

convergent

sequences

U

a sequence

: IN ~ ] N ,

is c o n t i n u o u s .

suppose

that

following

and a n ~ = lima n n-~o

i n E. T h e n t h e r e of r e a l s

of

form

the e x i s t e n c e It is t h e n Under

for the z e r o - n e i g h b o r h o o d s

w e can s h o w t h i s c o n t i n u i t y if we 2 g :~ ~ E of class C , using the

Lemma.

differential

a + l ~ h l + . . . + A n-n h

(x,h) ~ D n f ( x , h , . . . , h )

a denumerable

is e v e n of c l a s s C n, we

: IRn ~ E o f t h e

n t h e c o m p o s i t e m a p fog is o f c l a s s C , we g e t e a s i l y n D f ( x , - , . . . , - ) a n d its m u l t i l i n e a r i t y a n d s y m m e t r y . that the map

and then

symmetric.

for all m a D_s

that

(11'''"

and

exist

Recursively

I

n

exist

(cf.

to s h o w that E

( i . e . i f E is m e t r i z a b l e )

[7])

b 0 = limb n n-~o

a strictly

with a limit

enouqh

the asumption

fog is of c l a s s C n for all

lemma

'

of

:

be l i m i t s

increasing

of

function

10 = l i m ~ ,and a f u n c t i o n n n-x=

78

g

: ]R

2

~ E of c l a s s C°O s u c h t h a t

g(ln,~) This fact

it is f a l s e

3.

E metrizable :n

supposes

that

show that

n~2

. Then

could not

There

is

better

get t h e a b o v e

result

" , which

we

is o n l y a s l i g h t n

f has t h e p r o p e r t y

the norms.

However,

and this

that

but

between

if

of H a i n

[7].

to B a n a c h

classical

spaces)

.

notion

of

"

our notion

: E ~ L

n

C n a n d F r 4 c h e t - C n.

(E;F)

continuity

C n and

is in p a r t i c u l a r n+l

Proposition

Let

g

:

spaces

of

]Rn

(F' t h e t o p o l o g i c a l to a s s u m e

that

still weaker,

f(n)

true

~ Fr~chet-C n ~ C

the

~ F be s u c h t h a t d u a l of F) the

and

with respect

is n o r m - c o n t i n u o u s ,

if f is of c l a s s

coincides

spaces many

them yield

(n)

of f

then

C n+~.

to f

Hence

with the

different

same notion

"of

notions

class

log is of c l a s s C

suppose

Mackey-topology

t h a t F is l o c a l l y

classical

n+l

"of c l a s s

C~ ' '

as

f o r all

t h a t F is c o m p l e t e

(in f a c t

o f F is s e q u e n t i a l l y

complete,(cf

[9]

). T h e n

g is

C n.

Using and Siciak

our who

set-up

the proof

is a l m o s t

gave this proposition

in t h e

:

n

our C -notion

. For not-normable

[i0]).

or,

(in

spaces;

E of c l a s s C

"Fr@chet-C n

imply the

(cf.

complete,

difference

he u s e s t h e here

does not

the one we use

of c l a s s

call

~ F

all

it is e n o u g h

because

result

: IRn+l

: E×...xE

almost

4.

nA2

C~

(who r e s t r i c t s

for all g

shall

convex

C n if a n d o n l y

= Dnf

for B a n a c h

n o t i o n of F r ~ c h e t - C

exist;

locally

: ]Rn ~ E of c l a s s

C n, H a i n

if f is o f c l a s s

C shows

we a s s u m e

that

and this

is F r ~ c h e t - c n ;

proposition

, then the map

(n)

f(n) is c o n t i n u o u s ,

U {0}.

than the respective

f is of c l a s s

: E ~ F is of c l a s s C

1 6 F'

following

f is of c l a s s

f o g is of c l a s s C n+l

" m a p of c l a s s C n

Cn "

in t h e

F is o f c l a s s C n for all g

to

This

for all n £ ~

f : E ~ F be a m a p b e t w e e n

n 6~,

This proposition

If f

why

(n)

for n : i).

Let

;

In o r d e r

He

+ p. 5

also explains

Proposition

fog

= ao(n)

the

s a m e as t h a t g i v e n

c a s e n = i , cf.

[i].

by Bochnak

77

Combining

Boman's

following

theorem,

Theorem

1 .

theorem with propositions announced

in [5]

Let f : E ~ F be a m a p b e t w e e n

that E is m e t r i z a b l e are e q u i v a l e n t

and F

3 and 4 one gets easily the

:

(locally)

locally convex

complete.

T h e n the

spaces

following

and suppose conditions

:

i)

f is of class Cco

2)

f. (C~(IR,E))

3)

f*(C°°{F, IR)) C Cco(E, ]IR)

4)

f*(F')

= Cco( JR, F)

co

Corollary. smooth

For any F r ~ c h e t

structure

and the

m C

(E, IR) space E, the couple

on E. Hence F r @ c h e t

~-morphisms

between

Theorem

2.

of u n i t y

of~from

co

, C

can be c o n s i d e r e d

t h e m are e x a c t l y

If we w a n t to get o b j e c t s sure that p a r t i t i o n s

spaces

(~ (JR,E) the m a p s

(E, IR))

is a

as smooth

spaces

of class C°°.

Fr4chet manifolds

we m u s t m a k e

exist.

Let V be a p a r a c o m p a c t

space E w h i c h has the p r o p e r t y

Fr4chet manifold modelled

t h a t to each n e i g h b o r h o o d

over a F r ~ c h e t

V of zero there

co

exists

a C -function

Then

(JR,V),

between

C

(V, IR)) is a smooth

such spaces are exactly

Remark.

According

Fr4chet-manifolds C

f : E ~IR with

(V,W).

This

the u n i v e r s a l

V , W the n a t u r a l

property

structure

closedness

smooth

can be d e s c r i b e d that

= i and f(x]

= 0 for x ~ V.

on V. The

morphisms

the m a p s of class Cco.

to the. c a r t e s i a n

structure

f(0)

of ~ w e

structure

on the function

explicitly

for any such m a n i f o l d

get for any such space

in a simple way and has

X a Rap

f : X ~ C

(VrW]

is

co

of class C ~ C

(V,W)

iff ~

: X ~ V ~ W is of class C

again a m a n i f o l d

t h i n g s to look at in this vector

spaces

will give

set-up.

More natural

equipped with a compatible

some ideas

. One can then ask

? Of c o u r s e F r 4 c h e t m a n i f o l d s

in this direction.

smooth

: when

is

are not the n a t u r a l

are m a n i f o l d s structure~

modelled

The last

over

section

78

§4 C A L C U L U S

A smooth v e c t o r structure,

i.e.

an a r b i t r a r y

F O R S M O O T H V E C T O R SPACES

space

is a v e c t o r

such that the v e c t o r

object

X :

(S,C,F)

of ~ t h e

w i t h H(X, JR), is a smooth v e c t o r w a y an e q u i v a l e n c e f ~

relation

g ~=~ (f°e)" (0) =

space w i t h a c o m p a t i b l e

space o p e r a t i o n s function

space.

are

set F, being

If we define,

smooth

~-morphisms.

For

identified

for p 6 S, in the usual

"~ " on F by P (goc)" (0) for all c 6 C w i t h c(0)

= p

P then the q u o t i e n t space, tangent not,

is, due to c a r t e s i a n

c a l l e d the c o t a n g e n t

space of X at p as a q u o t i e n t

in general,

smooth v e c t o r evaluation remarks

a vector

space

seem v e r y useful

to o b t a i n

such a result,

E'

= p};

as a subspace

of F w i t h r e s p e c t

in this d i r e c t i o n

is the f o l l o w i n g

indefinitely

Let E be a smooth v e c t o r

{c 6 C; c(0)

here; way.

the

it is

into the

to the

but these It does not

on them.

we want to study spaces

also a smooth v e c t o r

we can introduce

spaces come in a natural

to put a t o p o l o g y

smooth v e c t o r

In o r d e r

of the space

but can be imbedded

formed by the d e r i v a t i o n s

smooth v e c t o r

The question between

space,

at p. We do not go further

show that

c l o s e d n e s s of ~ ,

space of X at p. S i m i l a r l y

differentiable

some r e s t r i c t i o n s

space,

: are the

(CE,F E)

~-morphisms

in the usual

on the spaces

its smooth

sense

seem useful.

structure.

We put

: = E* N F E

where E* notes the a l g e b r a i c real-valued

linear

Definition. points, cf

smooth

dual of E. So E'

functions

The smooth vector

generates

the smooth

is the v e c t o r

space of the

on E.

space E is c a l l e d c o n v e n i e n t

structure

and y i e l d s

if E' s e p a r a t e s

a comolete

bornology

on E;

[8]. This completeness

locally

convex t o p o l o g y

complete; Fr~chet

cf

condition

is e q u i v a l e n t

on E y i e l d i n g

[9]. A c c o r d i n g

space E the natural

E' as t o p o l o g i c a l

to the results smooth

to the c o n d i t i o n

of

structure

dual

t h a t any

is locally

§3 we see that for any (C

(JR,E)

, C

(E~ IR)) is

convenient. If c : IR ~ E is a space E

(i.e. c 6 CE)

hypothesis that

that there

~-morphism

one d e d u c e s exists

from LR to a c o n v e n i e n t

from the s e p a r a t i o n

a u n i q u e map,

denoted

smooth vector

and the c o m p l e t e n e s s

by c',

?

from]]{ to E such

79

loc"= F r o m the o t h e r asumption

(loc)"

(that E' generates the smooth structure of E) it

follows then i m m e d i a t e l y that e" we obtain

~O-morphisms

for all 1 6 E'

: ~ ~ E is also a

c (n) : ]R ~ E for n 6 ~

~morphism.

Inductively

and we see that c is indefini-

tely d i f f e r e n t i a b l e in the usual sense with respect to any locally

convex

t o p o l o g y on E y i e l d i n g E' as t o p o l o g i c a l dual. M o r e o v e r one v e r i f i e s that the (linear) map H(IR,E) ~ H(IR,E)

sending c into c" is a ~ - m o r p h i s m .

U s i n g this we get similar results for the general case T h e o r e m i.

Let d : E 1 ~ E 2 be a ~ Y - m o r p h i s m

:

b e t w e e n c o n v e n i e n t smooth vector

spaces. T h e n the map d~ defined by d~(a,h) ~-morphism.

=

: E1 ~ EI ~ E2

(~OCa,h)" (0) where Ca,h(l) = a + l h

For any a 6 E 1 the map d~(a,-)

is also a

is linear. The

H(E1,E2) ~ H(E 1 rl El, E2) sending ~ into d~ is also a If E~ separates points of E2, then o b v i o u s l y

(linear) map

~-morphism.

(H(E1,E2))'

separates

points of H(E1,E2). And if E~ g e n e r a t e s the smooth structure of E2~ then the remark at the end ~ ~(io~oc)

generate

' certainly

addition

that

H(EI,E2)

one sees

that

for

Theorem

2.

the

of the form

for 1 6 E~, c 6 C and ~0 a d i s t r i b u t i o n of compact support 2 E 1 ' smooth structure of H(E I, E2); since these functions are linear

the

(H(E1,E2))

of §2) the functions H(E1,E 2) ~

(cf

The

~-morphisms

generates satisfies

E 2 convenient

category is,

formed by

the the also

by

restriction

structure

of

H(E1,E2).

is

convenient.

completeness H(E1,

the of

E 2)

condition

convenient the

By

functor

smooth H,

showing

provided Hence

vector cartesian

in E 2 does,

we

have

spaces

:

with

closed.

Other p r o p e r t i e s of that category as well as the c a t e g o r y formed b y the same objects but with only the linear ~ - m o r D h i s m s

are being studied;

in

p a r t i c u l a r d u a l i t y and r e f l e x i v i t y questions. By i n t r o d u c i n g the spaces Ln(E1,E 2) of n - m u l t i l i n e a r

~morphisms

E l ~...~E i ~ E 2 one can of course introduce for a

~P-morDhism ~ ; E i ~ E 2 (n) between convenient smooth vector spaces the maps ~ : E] ~ Ln(EI,E2) w h i c h are also

~-morphisms,

derivatives (n)

and one has the usual relations between the higher

and the higher d i f f e r e n t i a l s dn~.

80

Added in proof. The convenient smooth vector spaces can be identified with the spaces considered by A. Kriegl ("Die richtigen R~ume f~r Analysis im unendlich-dimensionalen", preprint, Vienna 1981, to appear in Monatshefte fur Mathematik), namely the separated locally convex spaces which are

bornological and locally complete.

81

R E F E R E N C E S

[1]

J. B o c h n a k spaces"

2

3


Ac.

9 i0

Notes ii

12

A. Kriegl

XXI/4,

entre

espaces

lisses

engendr@es 1980,

D-

par des

367-375.

et v a r i @ t 6 s

de Fr4che~

p. 125-127. in V e c t o r

of smooth

77,

Spaces w i t h o u t

Norm",

1966.

1979,

p.

functions

d e f i n e d on a B a n a c h

63-67.

and functional

c o n v e x Spaces", Calculus

Springer

analysis",

Mathematics

F.W.

S.H.

Schanuel

Teubner

1981.

in locally

convex

spaces",

Lecture

1974.

glatter

1980.

Mannigfaltigkeiten

and W.R.

Zame

und v e k o r b ~ n d e l " ~

: ~'On C°° F u n c t i o n

Spaces",

1981.

N. V a n Oue and G. Reyes de Whitney",

Recherches

appl.

der F e r n u n i v e r s i t ~ t

1977.

Wien

U. S e i p

diff.

: "Bornologies

: "Eine T h e o r i e

Lawvere,

249-268.

abgeschlossene

Math.

et G4om.

Soe.

Dissertation,

tension

14

Am. Math.

417,

1967, p.

kartesisch

ferm@es

: "Calculus

:"Differential

in Math.

20,

car4siennement

30, S p r i n g e r

26, N o r t h - H o l l a n d

Keller

Preprint 13

in Math.

: "Locally

vector

and of its c o m p o s i t i o n s

Scand.

aus d e m Fachber.

: "A c h a r a c t e r i z a t i o n

H. J a r c h o w

Math.

erzeugte

1981,

and W. B u c h e r

H. H o g b e - N l e n d

H.H.

de Top.

Paris 293,

Proc.

in t o m o l o g i c a l

.

: "Applications

Sci.

Hain

Studies

7-48

Cahiers


R.M.

p.

: "Categories


space", 8

of a function

Seminarberichte

5, 1979,

functions

p. 77-112

of one variable",

Kategorien",

L e c t u r e Notes 7

39, 1971,

: "Dutch M o n o i d e

C.R. 6

"Analytic

: "Differentiability

functions

monofdes", 5

:


Hagen 4

Studia Math.

J. Boman with

and J. Siciak

DMS 80-12,

21,

1981,

des d i s t r i b u t i o n s

8, G4om.

Universit4

: "A c o n v e n i e n t

Algebra

: "Th6orie

Expos@

Settina

diff.

synth,

de M o n t r @ a l

et th@or6~nes d'ex~

fasc.

2, R a p p o r t

de

1980.

for S m o o t h Manifolds".

J. of p u r e and

p. 279-305. S e c t i o n de M a t h @ m a t i q u e s U n i v e r s i t ~ de Gen@ve 2-4, rue du Li6vre CH~I211

GENEVE

24

E n r i c h e d algebras,

spectra and h o m o t o p y limits

John W. Gray O. Introduction.

The purpose of this paper is the same as that of

[5];

to show how certain p r o p e r t i e s of h o m o t o p y limits are consequences of w h a t either are or should be standard facts about categories e n r i c h e d in a closed category. in

The p r o p e r t y to be e x p l a i n e d here is as follows:

[16], T h o m a s o n shows that the d e g r e e w i s e h o m o t o p y limit of a d i a g r a m

of p o i n t e d simplicial spectra is a pointed simplicial spectrum. this the h o m o t o p y limit in the category of such spectra. reasonable to suppose that, category,

in fact,

it is the h o m o t o p y limit in this

but two things have to be proved,

of p o i n t e d simplicial

i) .

The category

Spec K,

spectra is a complete simplicial category,

only such categories have h o m o t o p y jections

He calls

It is e m i n a n t l y

pr n : Spec K, ÷ K,,

e n r i c h e d left adjoints,

limits,

ii) .

for each degree

since

The component pro-

n,

and hence preserve h o m o t o p y

have s i m p l i c i a l ! y limits.

The r e q u i r e d tools are m o s t l y at hand for o r d i n a r y categories in the form of known p r o p e r t i e s of the category for an e n d o f u n c t o r

S

of a category

A.

Dyn S

of algebras

In Section 1 these tools are

s h a r p e n e d and e x t e n d e d to the case of e n r i c h e d categories. Spec A

In Section 2

is d e s c r i b e d for an arbitrary complete V - c a t e g o r y

category)

and a pair of V - a d j o i n t

functors

Z--4Q.

(V

a closed

Finally,

in Section

these results are s p e c i a l i z e d to pointed simplicial spectra.

Note that

the spectra treated here are those for w h i c h phism.

i.

V-categories.

cocomplete,

T h r o u g h o u t this section

symmetric, m o n o i d a l

X n + ~Xn+ 1

V

denotes a complete,

closed category.

category of V - e n r i c h e d categories

is an isomor-

and functors,

V-cat

denotes the

regarded both as a

symmetric, m o n o i d a l closed category itself and as a 2 - c a t e g o r y in w h i c h the 2-cells are V - n a t u r a l transformations; t : F ~> G : A ÷ B the diagrams

between V - f u n c t o r s

A(A,B)

such that for all

commute.

A

and

B,

FA'B > B(FA,FB)

GA,B I B(GA,GB)

i.e., natural t r a n s f o r m a t i o n s

](l'tB) (ti,l)> B(FA,GB)

For basic information,

see

[5],

[8] and references therein.

3,

83

i.i.

Proposition.

Proof:

It is w e l l

to s h o w t h a t If

V-cat

it h a s

A • V-cat

phisms

in

If

• V

that

V-cat

cotensors

then

V.

2 ~ A(f,f')

known

is a c o m p l e t e

2 ~ A

with

2-category.

has

the

limits.

arrow

category

is the V - c a t e g o r y

f : A + B

and

Thus

whose

f'

: A' ÷ B'

d1 - - >

A(B,B')

it is s u f f i c i e n t 2

(cf.

objects

[21]).

are m o r -

are t w o such,

then

is the p u l l b a c k 2 ~ A(f,f')

(f,l)

A(A,A') It is e a s i l y

checked

transformations

t

that

8~ = t

that

8f = f.

1.2.

Proposition.

that

there

: F ~>

where

Q

(i, f,)-> A ( A , B ' )

G

is a n a t u r a l

: A ÷ B

: do ÷ d I

bijection

and V-functors

between

~

in the V - n a t u r a l

V-natural

: A ÷ 2 ~ B

transformation

such such

Let K B

>

B'

Ii be a d i a g r a m F'--4U' tion

~

of V - f u n c t o r s

Then

there

: H U :> U ' K

A

--->

A'

--

H

--

such

that

is a n a t u r a l and

there

are V - a d j u n c t i o n s

bijection

8 # : F ' H =>

KF

between

such

that

F--~U,

V-natural for all

transformaA

and

B

the d i a g r a m s H A(A, UB)

D

:

> A' (HA,HUB)

(I'SB) - - - >

A' (HA,U'KB)

f J

f B_(FA,B)

> B'(KFA,KB) _

- 0-# - i) >

B' (F'HA,KF)

( A' commute Proof: n

(cf. Given

: A =>

the that

[19]) .

UF,

8 : H U ÷ U'K, s : F U => B,

adjunction this

natural

establishes

~'

then

0 # = E'KF

: A' => U ' F ' ,

transformations. a bijection

See

o F'SF

and [4],

as i n d i c a t e d .

s' I, The

o F'HN, : F'U' 6.6

where

÷ _B'

are

for the p r o o f

diagram

D

commutes

84

because bottom

of the are

commute

commutativity

the s i d e s ,

top

of F i g u r e

and bottom

b y the d e f i n i t i o n

1 in w h i c h

of

D.

of V - n a t u r a l i t y .

The The

the

sides,

regions other

top

and

labeled

regions

*

commute

trivially. 1.3.

Definition:

category

S + A

: S A ÷ B.

i)

Let

S

: A ÷ A

is the V - c a t e g o r y

If

~'

: SA'

+ B'

S + A(~,~')

be

whose

then

Pl

+ S

has

The

comma

are m o r p h i s m s E V

is the p u l l b a c k

2 ~ A ( { ,~ ') --

d o

S

A(A,A') A

objects

S + A({,~')

poI Dually

a V-endofunctor.

as o b j e c t s

A(SA,SA')

morphisms

{

: A ÷ SB

and

A + S(~,~')

w


of the

diagram A + S(~,~')

.oL

Pl

objects of the

Dyn

S

denotes

are m o r p h i s m s

~

1

d1

2 ~ A(~,~') ii)

> A(B,B')

> A(SB,SB')

the V - c a t e g o r y

: SA + A

and

of S - a l g e b r a s

D y n S(~,W')

--

Dually, objects

coDyn

S

denotes


equalizer

of the

~

dlPl

Its

is the e q u a l i z e r

~ A(A,A')

do

the V - c a t e g o r y

: A + SA

and

of S - c o a l g e b r a s coDyn

S(~,~')

two m o r p h i s m s Pl % S(~,~')

A(A,A') d0P 0

1.4.

A.

two morphisms S + A(~,~')

(cf.

e V

in

[6] a n d Remarks:

[i0]) . There

are d i a g r a m s S +A

A_--K--> A_

A+S

A_ ~ >

A_

6 V

in

A. is the

Its

85

-

W v

~

A

A

A

~

v

r~

v

~

r~

A

A

T~I

A

v

! r~ ~J

~D

,r-I ~J

r-~

v

A

f~

.

F 2

S

coDyn

& --K-> & which

are

universal

transformation 1.5.

for

SF =>

Proposition.

isomorphisms Proof.

The

V-functor s

: X ÷ A

F 1 =>SF2)

.

and

a V-

Similarly

there

diagrams

Dyn

S

A

a single

F

Given

s

V-functor

F

F ~>

over

A

transformation A

determines

~ T s

equations

: A

are

and

: A :>

that

¢ T ÷

: X + A (cf.

~

TS

S

there =

• nU I. that

--

~ = F

--i

are

coDyn

determines

such

to

a V-natural

6.4)

then

Dyn

= T9

S ¢ A _

equivalent

and

[6]

S --J T

x A ~

such

S

A

SF)

a V-adjunction

¢ T

: S + A ÷

adjunction

A_

(resp.,

¢ A = A

natural ~

: ST =>

The

FI,F 2

(resp.

T

Vover

A

a unique Similarly,

8~

= sU 2

since,

' S~.

for

instance @S ~ = (sU 2 =cU 2 =8 so

s

n

Clearly

id.

=

• S~)~ • S(Te

• aSU 1

~

and

A

is

~

=

eU2~

• S~

• ~U I)

= ~U 2

SnU 1 =

e •

restrict

to

• ST@

(~S

• S~U 1

• Sn)U 1 =

give

the

@

second

isomor-

phism. 1.6.

It

is

created

by

U

limits.

a complete

well

Recall

indexed

known

: Dyn that

limit

that

S ÷ A if

{G,F}

Dyn

(cf.

Then

the

moment,

there

is

suppose

G

: I ÷ V A

a diagram

for like

S

then

ordinary

limits

has

[i]) .

e

Dyn

S

is

We

and

show F

here

: I ÷ A

that are

which A

has

a

are indexed

V-functorsthen

satisfies

A(A,{G,F}) For

V-category

V-category.

Proof.

the

If

Proposition.

complete

~

[I,V] ( G , A ( A , F ) )

clarity the

one

that in

S 1.2,

: A ÷

is

a V-functor.

87

AoP

S °p

'l

{-,F}

A(-,F)

and hence

t

a bijection

=> B ( - , S F )

and

: A(-,F) = > B ( - , S F )

tA

: A(A,F)

Then

t

cG

° S

÷ B(SA,SF)

corresponds

between

functors

: S({G,F})

÷

with

S

be t h e

~

Dyn

claim S.

that Then

[I,V]

o {-,F}

transformation

Let whose

BoP

components

tA,i = S A , F i

transformation whose

: A(A,Fi)

c : {-,SF}

components

to an e n d o f u n c t o r to a f u n c t o r

: SF = > F .

=>

÷B(SA,SFi) S o {-,F}


S F

: A ÷ A.

: I ÷ A

A V-functor

and

a V-natural

Let

I = {G,~} 6 D y n there

B(-,SF)

in

i = {G,~}

We

]

transformations

=> S

components

codomain

corresponds

transformation

V-natural

have

{G,SF}

>

{-,SF}

to a n a t u r a l

Now we return : I ÷ Dyn

id

between

o S

BoP

{-,SF}

[I,V]

A(-,F)

>

o cG S.

: S{G,F} ÷{G,F} Let

~

: SA + A

be a n o b j e c t

of

is an e q u a l i z e r d i a g r a m

Dyn S(~,I)

--> A ( A , { G , F } )

> A(SA,S{G,F) }

t (l'CG) A ( S A , {G,SF})

J (1,{l,~})

[

A(SA, {G,F}) which

is i s o m o r p h i c

to an e q u a l i z e r d i a g r a m (i, t A)

E-->

[I,V] (G,A(A,F)

(i,

>

[I,V] ( G , A ( S A , S F ) )

(y.i i ~ " ~

(i, (i,~)) [I,V] (G,A(SA,F))

the

crucial

[I,V]

one

step being has

given

by the

commutative

diagram

an equalizerdiagram A(~,~)

--> A(A,F)

tA --->

A(SA,SF)

A(SA,F)

in 1.2.

But

in

88

and

[I,V] (G,-)

preserves Dyn

Hence

I = {G,~}.

1.7.

Proposition.

ates

equalizes

so

S(tg,l)

= E -- [I,V] (G,A(%,~))

S

preserves

If

: A ÷ A

coproducts

S

then

gener-

a free V - m o n a d .

Proof.

See

monad

(triple)

is o b v i o u s to h a v e

[i],

[3],

[9] , [12]

problem.

This

that

them

one

gets

value

on o b j e c t s

whose

V-structure

m

is the

is

S(A)

for d i s c u s s i o n s

simplest We

example.

Let

possible

sketch S

:

~m

A(Sr~A,SmB)

of the case

the d e t a i l s

: A ÷ A

I ] Sn(A) , n=0 by the c o m p o s i t i o n

is g i v e n

m > <SA'~B > ~-

A(A,B)

[16]

a V-monad.

in the m a i n

whose

and

free

in w h i c h

it

in o r d e r

be the V - f u n c t o r

where

S0(A)

> ~-

~ (SmA,

= A,

and

(l,in m)

m

~(

I I SnB)

n

tm I SmA, In I

Sng)

/I

where and

: SmB :>

SB

in the

m'th

s = in I : S ~>

in m

S.

Define

~

whose

components

are

that

formations

~A

° inm,n

and

S =

is an S - a l g e b r a , valent and

induces 1.8. U

:_ I I

I I smsnA m=On=O

= inm+n"

(S,~,~)

then

the

to the r e c u r s i v e

kn+l

= Ii

°Sln

S + A

has

S

>

Let

D = in 0

: Id =>

to be the t r a n s f o r m a t i o n

~,~

algebra

and

: A ÷ A

If

equations

s

I =

preserves given

s

coproducts

[I P

seen

Hence,

: AS ÷ Dyn

by

:

1 1 : SA ÷ A

Im+n = k m o S m l n ) .

adjoint

are V - n a t u r a l

are e a s i l y

I0 = id,

a V-isomorphism

a V-left

I I sPA p=O

is a V_ - m o n a d .

(actually,

If

Then

conditions:

(by c o m p o s i t i o n )

Corollary.

: Dyn

S

the m a p s DA

such

summand.

: S S =>

S. then

trans-

sPA ÷ A

to be e q u i arbitrary, s

: S =>

89 oo

F(A) Proof:

A~ ÷ A

Composing

has

with

s

=

(fA

a I-left gives

:

co

[ I SnA n=l

adjoint

the m a p

given

fA

:

I I SnA) n=0

by

Fs(A)

: S S A ÷ SA

co

fA

>

= ~A

co

[I n=0

SSnA

>

II n=0

SnA co

satisfies

fA o in I n = inn+l"

Rewritting

for

1.9.

Definition.

coDyn

S

1.10.

Let

If

S

The

inclusion

in

= K(~)

the m a p s SA = lim A

qn+l

° Sn~

given

by

=

] ] SnA, n=l

then

has

coDyn

subcategory

sequential

colimits,

of

coDyn

S.

a V-left

adjoint

K

S

is the

- - ~

- - 9

n

n

sequential let

qn

of

A --~> SA. then

whose

value

on

isomorphism.

: l i m SnA ---~> l i m s n + I A ~ S l i m

in the and

the V - f u l l

~:

subcategory

functor

~: A ÷ SA

Let

denote

preserves

is a V - r e f l e c t i v e

an o b j e c t

where

coDyniS

by i s o m o r p h i s m s

Proposition.

Proof:

SSnA

n > i.

determined

coDyniS

co

If n=0

'

fA ° inn = inn

: S S A ÷ SA.

in w h i c h

SnA >

n

colimit

are

: s n A ÷ SA

given

by

be the m a p

sn~ : S n A ÷ s n + I A .

to the

colimit,

so

n = qn"

The maps

sn+IA

that

=

qn+l

induce

the two

sn+IA

=

qn+l

lim s P A

< -

isomorphisms

h n = hqn

I o qn+l

Sqn

lim s P + I A

B

L

Sh > SB

SSA commutes

are

>

= Sqn"

: S n A ÷ B.

~

S SnA

>

Hence,

in

~ o h o qn+l

= Sh

I

o I o qn+l

iff

~ o hn+ 1 = Sh ° Sq n = S h n.

90

Step

1.

To s h o w

also

the

square

that

K

is left

adjoint

h0 A - -

4

is an i s o m o r p h i s m For

suppose

In p a r t i c u l a r , follows

that

Conversely, sively

by

then

square

square

hl = 4 -1

(~)

one

i.e.,

that

h0

h n + l = 4 -1

h = n 2.

making o Sh

n

.

square

K

o Sh n o s n ¢ =

makes

is V - l e f t

step.

adjoint

We m u s t

also

is an i s o m o r p h i s m , s(sn~,4)

6 V)

square

E --> A(SA,B) II lim _ A(SnA,B) A ( s n + I A , B )

is the e q u a l i z e r ( w h e r e

-> A(SSA,SB) II

S > lira
SB

En ÷ A(SnA,B)_ the

h I o ¢ = h0, (0)

..... > B

can be t a k e n A(sn+IA,sB)

For

(0)

14

sn+IA 4

has

square

consider

sn~[

(n)

also

square

hn+ 1 = 4 -1 o Sh n.

= hn-

square

SnA

coDyn

(0)

iff

Then

Then

= 4 -1

: SA ÷ B

Showing

of the r e c u r s i v e

If

commutes

Since

= 4 -I o S(hn_l)

Step

(~)

commutes.

o Sh0.

hn+ 1 ° sn~

Hence

SB

Sh 0 o ~ = 4 o h0; given

consider

I Sh 0 SA - -

If

inclusion,

B

(o)

commutes.

to the

(1'4-1)

...>. A(SSA,B) tl

> _A(SnA,B) id (I,i)

is omitted) >

A(SA,B) iJ

A($n+IA,sB) (l'4-1)>- lJ_mA(sn+IA,B)

n+l

map.

map

is the

Ei

third

is

and

En+ I. map,

by the

is also

the

diagram (i,~ -I) > A ( S n + I A , B )

~ - - > (i,~ -I)

A(sn+IA,B)

/

S.

is the

coDyn

S

where

SF,

consider

A construction indexed

then

~

limit

very

of

G

and

is an i s o m o r p h i s m

{G,~}.

Let

of V - a d j o i n t topological

1.6

and

transformation

= S{G,F}

S.

indexed

isomorphisms,

complete.

{G,SF}

so is

V-spectra.

A

be a c o m p l e t e

endofunctors situation.)

for

A = < A 0 , A I, • ..,>.

n ~ 0. Let

Prn(-) + = P r n _ 1 A) ;

S

to a V - n a t u r a l ÷

the

of

are

V-function

to that

and hence

that

system

If

are

corresponds {G,~}

2.

inverse

Proposition.

and

pair

K

in the

Let

A.

V-category

: A_~ ÷ _ Am n ~ 1

and

and

(The n o t a t i o n

_ A ~ = --~ A n=0 A n = Prn(A)

Write

(-)+ for

of

be

with and,

PRO(-) + = ~

+ =

Z--~ ~

be

a

to r e f l e c t

projection

V-functors

sometimes,

the u n i q u e

i .e. ,

let

is c h o s e n

V-functor

such

(the i n i t i a l

object

92

÷

÷

4r~

I

A

A

A A

,-4

÷

4-

r./l

t"xl +

"F

A

,~ +

I A

A

~

~

÷

~-

I

2

5

m

r.~

2 c-1

,~ +

+

÷

2 ©

v I A

A

A v

I

U?

r~

2

~

+

4-

4-

,~

v I

÷

A

~

A

93

Similarly

(-)

: A~

_

÷

satisfies

A ~

Clearly

(-)+---~ (-)-

Prn(-)-

= Prn+ 1

so

=

is a V - a d j u n c t i o n .

Now

E

and

~

co

endofunctors

of

A ~,

g i v e n by

Z~ = ~ n=0 co

V_-adjoint.

Hence

Z+ =

~

and

-

(-)+ o Z and

also i n d u c e

oo

~

oo

Q

= --~- ~, n=0

which

are

_

= Q

o (-)

are V - a d j o i n t .

Here Z+ 2.1.

Definitions.

subcategory

of

i)

Prespec

Prespec

A

= =

A = Dyn Z +.

corresponding

ii)

to

Spee A

--

isomorphism 2.2. of

Dyn Z+ =

Remark. Aco

together with

u n d e r the

1

c o D y n Q-

An o b j e c t

is the V - f u l l

coDyn. Q-

of

of 1.5.

Prespec A

a map

~

is an o b j e c t

: Z+A ÷ A

whose

A = < A 0 , A 1 .... >

components

are m a p s

~ n + l : EAn ÷ An+l" It b e l o n g s to S p e c _A if the t r a n s p o s e # ~ n : An ÷ ~ A n + l are i s o m o r p h i s m s for n ~ 0.

maps

2.3.

Spec A

Theorem.

Prespec

A

If

~

preserves

are c o m p l e t e

sequential

V-categories

colimits

and there

then

are p a i r s

and

of V - a d j o i n t

functors K Spec A

A~ < --

U complete. such that

The

A

>

__

Prn

left a d j o i n t

(Ln(A)) p = A

if

Ln

is the

p = n

Z-

and

otherwise. ii) adjoint

By 1.6,

it p r e s e r v e s

Prespec

coproducts

A

is c o m p l e t e .

apply,

g i v i n g a left a d j o i n t F such that n UF(A) = I I zPA and the s t r u c t u r e m a p n p=0 n-p c o m p o n e n t s the m a p s n n n+l Z( p=0 I I ZPAn_ p) = p=0 I I zP+IA n-p = I I q=l which

omit

so

~

has a r i g h t

of 1.7 and 1.8

UF(A)=

I I (z+)P(A) . Thus p=o ~UF(A) ÷ UF(A) has as

n+l

÷ I I EqAn+l-q ~qAn+l-q q=0

the f i r s t summand. ii)

limits

Since

so the c o n s t r u c t i o n s

Since

Spec A =

~

has

coDyni~

a V-left

adjoint,

is c o m p l e t e ,

it p r e s e r v e s

by i.ii.

indexed

By h y p o t h e s i s ,

94

preserves

sequential

K

: Prespec

A prespectrum

~:

colimits,

A =Dyn

~+A ÷ A

so 1.10

Z+ ~

applies,

CODyn

corresponds

~- ÷

to

giving

coDyni~-

a left

~ Spec

a coalgebra

~#:A ÷ ~-A

#

components

: An ÷ ~An+l"

A-object

the

The

colimit

reflection

lim(~-)n(A)

into

.

eoDyni~

Thus

has

UK(A) n = l i m

>

The structure

map

UK(A) n = l i m ~ 3 A n + 2.4.

j ~ lim

the

transpose

components

a]+lAn+j+ 1 ~a

i)

The

the

the

isomorphism

isomorphisms

l i r a ~ 3 A n + l + j = a UK(A)n+ 1 >

composition

in h o m o t o p y

FL 0 (A) n ~ 2nA

of

are

>

Remarks.

of i n t e r e s t

is

whose

.

~3An+ j .

j

ZUK(A) ÷ UK(N)

# a-lim(a-)n(A)

as

- - >

n

lim(a-)n(A)

with

-

~n

underlying

adjoint

theory

Q~ = K F L 0

(cf.

[i1]) , is

: A ÷ Spec

given

as

A,

which

is

follows:

so Q~(A) n = lira ~ J ~ n + J A 3

In p a r t i c u l a r ,

Q(A)

= Q~(A) 0 = lim ~ J z J A

is the

stabilizing

functor.

>

ii) indexed H

Let

limits

: I ÷ Spec

Pn = P r n U I

since

A,

all

let

: Spec

three

A ÷ A.

functors

Then

Pn

have

V-left G

H n = pn H.

Then

for any

spectra.

Let

K =

preserves

adjoints.

: I ÷ V,

If

one has

{G,F} n = {G,Fn}.

3.

Pointed

tesian

closed

category with

a map K,

such

that

U

1.6,

x

the

= 1

sets,

and

i.e.,

objects

1 : K ÷ K

denote

sets;

has

a left

follow

coproducts)

Let

(the t e r m i n a l

is c o m p l e t e

not

[A°P,sets] let

denote K, X

the

denote of

K

car-

the together

: 1 ÷ X.

I(X)

K,

does

preserve

of s i m p l i c i a l

simplicial

as a K - c a t e g o r y .

: K, ÷ K

this

category

of p o i n t e d

3.1.

By

simplicial

object

as a K - c a t e g o r y . adjoint

F

given

construction

From

general

.

the

constant

of

K) .

The

underlying

by

f r o m the

the

of

Then

F(X)

= X ~

1.8

since

description

of

K-functor

K, = D y n

i.

functor i,

(although

i(-) D y n S,

does

following: i)

If

Y,,Z,

e K,,

then

K,(Y,,Z,)

e K

not

one has


95

- - >

K.(Y,,Z,)

1 = K(l,l)

z[ [(l,z) K_(Y,Z)

- - > (y,l)

ii)

X E K,

then

X ~ Y, 6 K,

1 = K_(X,I) iii)

K,

Z : K(I,Z)

is a l s o

is g i v e n

(l,y~>

tensored

K_(X,Y)

over

X × 1

>

by

being

K, X ~ Y,

the p u s h o u t

1

[ X × Y iv)

One has

the

following

a) K . ( X ~ Y.,Z.) b)

correspond Further,

I,Y,)

as a c l o s e d

K,

X, A

to

If

<X x y , x

X,,Y,

>

Y

> X, A

K,

is a s y m m e t r i c

monoidal

by

1 + K,(X,,Y.)

and ®-product

press

the

UK,(X,,Y,) normal

÷ K(X,Y)

and

closed

functor.

products

(i.e.,

functor.

Hence d)

-al

e)

U

(X ~

closed

equipped UX,

The

(Y ~

preserves preserves

with

and

ii) .

1

adjoint i)

=

by

Y,

with

internal

1 + X,A

hereafter.)

The

Y..

hom (We s u p -

underlying

canonical m a p s

× UY, ÷ U ( X ,

left

i) A

category

given

f r o m the n o t a t i o n comes

a)

x y>

given

: K, ÷ K

1 ÷ K,(X,,Y,)

( f r o m iv)

(i x y)

Then

base points

Let

the p u s h o u t

X

U

e K,,

X, ÷ 1 ÷ 1 ~ Y,

denote

(X x i) "

functor

~ Z.)

@ Y,,Z,) 0

category.

Y.

~ K.(Y.,X

= K(X,Y) .

by adjointness

let

formulas

~ K(X,K.(Y.,Z.))

K,(X,,Y,) n = K,(A[n]

c) K , ( X ~ 3.2.

> X @Y,

× Y,) F =

making

(-) ~

(X × Y) ~

i)

1

to

U

into U

a

preserves

so it is a l s o

a closed

tensors;

i.e.,

(X × Y) m

1 = X @(Y ~

cotensors;

i.e.,

U ( X ~ Y,)

= K(X,Y) .

i)

96

3.3.

Definitions.

i)

Let

S1

be the c o e q u a l i z e r

(in

K)

of the two

maps do A[0] _ _ . ~ dI S,1

and let

If

A

3.4.

tially K . - s m a l l

1 S,

~A = S, ~X E A

: A ÷ A

If

A

objects,

in

then

are defined and are K,-

is called s e q u e n t i a l l ~ K , - s m a l l if

has a s t r o n g l y g e n e r a t i n g

then

S2A

preserves

A,

colimits.

K,.

Hence.

if

{A m}

then

1 1 A(Aa,S , ~ lim B i) -~ K,(S..A(Aa, - - >

lim B i) - - >

= lim K,(SI,A(Aa.Bi))

-~ lim A(A

- - >

-~ A ( A

family of sequen-

sequential

is clearly s e q u e n t i a l l y K , - s m a l l in

is the f a m i l y

K,-category.

sequential colimits.

Proposition.

Proof:

and

An object

perserves

SI

>

is a complete and cocomplete

ZA = S ~ - : A ÷ A adjoint. iii)

q

qd 0 : 1 = A[0] ÷ S 1

have the base point

ii)

A(X,-)

A[I]

,S 1 ~ B i)

>

1 , lim S, ~ B i) - - >

so

S

~ lim B. = lim S, ~ B. _ _ >

3.5.

l

Example:

_ _ >

i "

{A[n] m

i}

tially K, small objects in

is a strongly g e n e r a t i n g family of sequenK,.

However.

~K

(-) = K,(S~.-)

preserves

__W

sequential colimits anyway. 3.6.

Definition.

If

A

is a K. -category.

with the same objects and By category,

[5], 2.2.3, if

then

Furthermore, limits in

U,A

by

A

given then,

U,A(A.B)

is a complete and/or cocomplete K,-

U,A.

there is a close r e l a t i o n s h i p b e t w e e n indexed if

H F

Namely,

the K - f u n c t o r s

spond to K , - f u n c t o r s

is the K - c a t e g o r y

= U(A(A.B)) .

r e g a r d i n g the left adjoint

F : K ÷ U,K,.

U.A

is a complete and/or cocomplete K-category.

[5], 2.4.3,

and in

A

then

H

: I ÷ U,A to

: I ÷ H,A

H # : F,I ÷ A

and

and

U

and

G : I ÷ K

are

as a K - f u n c t o r FG

: I ÷ U,K,

(FG) # : F . I ÷ K,

corre-

and

{G,H}u, ~ = { (FG)#,H#}A. If is defined in

~

is any K-category,

then an indexing functor

[5], 4.5.1, which reduces to

free K - c a t e g o r y on an o r d i n a r y category,

N(I/-)

in case

Z I :I ÷ ~ --is the

such that indexed limits over

97

ZI

are homotopy

limits in

U,A i)

limits.

Hence

for a complete K , - c a t e g o r y

holim H = {ZI,H}u,A(= --

ii)

fiN(I/i) )

--

holim H = { ( Z I ~

-

'

the ¢otensore

sors in 3.7.

U,A

(cf.,

Theorem.

generating

If

are actually

A

functors

Example:

by 2.3,

If

S : k-sp ÷ K

of coten-

Pn

k-sp

denotes

is given as follows:

k-sp(X,Y)

= S(Y x) , Z @ X = IZI × X of pointed k-spaces

if

which has a strongly

objects,

has homotopy

then

Spec A

is

limits that are pre-

: U, Spec A ÷ U,A, n ~ 0. and 3.4.

of k-spaces

1 - I : K ÷ k-sp

structure category

K,-small

2.4, ii),

The category

K-category.

ization and

A

is a complete K , - c a t e g o r y

served by the projection

3.8.

i) ~ H#(i))

i

the same by the construction

U, Spec A

cocomplete

H(i))

[5], 2.2.3).

family of sequentially

Immediate,

~

I)#,H#}A(= I (N(!/i)//

a complete K,-category.

Proof:

homotopy

U.A

--

where

A,

are given by either of the formulas

the singular X, Y • k-sp and

is a complete

denotes

functor, and

Z ~ X = X IZl

is a complete

and

geometric

real-

then this

Z • K,

then

Similarly,

the

and cocomplete K, category.

98

Bibliography [i]

M. Barr, C o e q u a l i z e r s

[2]

A. K. B o u s f i e l d and D. M. Kan, H o m o t o p y limits, Localizations, New York,

[3]

and free triples, Math.

Lecture Notes

in M a t h e m a t i c s

Z. i16(1970),

Completions and

304, Springer-Verlag,

1972.

G. Gierz, K. H. Hofmann, K. Keimel,

J. D. Lawson, M. M i s l o v e and

D. S. Scott, A C o m p e n d i u m of Continuous Lattices, New York, [4]

J. W. Gray, Formal Category Theory:

J. W. Gray,

Adjointness

for 2-categories,

391, Springer-Verlag, New York,

Closed categories,

Pure and A p p l i e d Alg. [6]

Springer-Verlag,

1980.

Lecture Notes in M a t h e m a t i c s [5]

307-322.

lax limits, and homotopy

19(1980),

1974.

limits, J.

]27-158.

J. W. Gray, The e x i s t e n c e and c o n s t r u c t i o n of lax limits, Cahiers de Top. et G~om. Diff.

21(1980),

277-304.

[7]

J. W. Gray, Two results on h o m o t o p y

[8]

G.M.

limits,

(to appear).

Kelly, The Basic Concepts of E n r i c h e d C a t e g o r y Theory,

(to appear). [9]

J. Lambek and B. A. Rattray, Trans.

[10]

Am. Math.

L o c a l i z a t i o n and sheaf reflectors,

210(1975),

E. Manes, A l g e b r a i c Theories, Springer-Verlag,

[ii]

Soc.

J. P. May,

279-293.

Graduate Texts in Mathematics,

1976.

Infinite

loop space theory, Bull. Am. Math.

Soc.

83(1977),

456-494. [12]

D. Scott, The lattice of flow diagrams, A l g o r i t h m i c Languages, 188, Springer-Verlag,

[13]

D. S. Scott,

E. Engeler, New York

Continuous

(1971),

lattices,

[14]

1972,

311-366.

in Toposes,

and Logic, Lecture Notes in M a t h e m a t i c s York,

S y m p o s i u m on Semantics of

ed., Lecture Notes in M a t h e m a t i c s

Algebraic Geometry


New

97-136.

M. B. Smyth and G. D. Plotkin, The c a t e g o r y - t h e o r e t i c solution of recursive domain equations, Edinburgh,

1978.

D. A. I. R e s e a r c h Report No.

60,

99

[15]

R. W. Thomason, gories, Math.

[16]

H o m o t o p y colimits in the category of small cate-

Proc. Camb. Phil.

R. W. Thomason,

Soc.

85(1979),

91-109.

A l g e b r a i c K - t h e o r y and etale cohomology,

Preprint

1980. [17]

H. Wolff, Free monads and the o r t h o g o n a l s u b c a t e g o r y problem, Pure and A p p l i e d Alg.

[18]

M. Tierney,

13(1978),

[19]

233-242.

C a t e g o r i c a l Constructions

A seminar given at the ETH. Mathematics

ZUrich,


G. M. Kelly, A d j u n c t i o n

in Stable H o m o t o p y Theory.

in 1967. Lecture Notes in

New York,

1969.

for enriched categories,

Reports of the

Midwest Category Seminar III, Lecture Notes in M a t h e m a t i c s Springer-Verlag, [20]

1969,

106,

166-177.

L. G. Lewis, The stable category and g e n e r a l i z e d Thom spectra, Dissertation,

[21]

New York,

U n i v e r s i t y of Chicago,

R. Street, F i b r a t i o n s G6om. Diff.

21

(1980),

1978.

in b i c a t e g o r i e s , 111-160.

J.

Cahiers de Top. et

GENERAL

CONSTRUCTION

IN T O P O L O G I C A L

OF M O N O I D A L

, UNIFORM

Georg

CLOSED

STRUCTURES

AND NEARNESS

SPACES

Greve

Abstract/Introduction: In the f o l l o w i n g p a p e r we c o n s i d e r t o p o l o g i c a l s t r u c t u r e s on f u n c t i o n s p a c e s and c a r t e s i a n p r o d u c t s b e i n g c o n n e c t e d by an e x p o n e n t i a l law of the f o r m C(XeY,Z) ~ C ( X , C ( Y , Z ) ) . T o p o l o g i c a l c a t e g o r i e s p r o v i d e d w i t h s u c h a " m o n o i d a l closed" s t r u c t u r e are s u i t a b l e b a s e c a t e g o r i e s for t o p o l o g i c a l a l g e b r a , a l g e b r a i c t o p o l o g y , a u t o m a t a - or d u a l i t y t h e o r y , in p a r t i c u l a r if ~ is s y m m e t r i c or the u s u a l d i r e c t p r o d u c t . We s t a r t f r o m a p u r e l y c a t e g o r i c a l p o i n t of v i e w p r o v i n g an e x t e n s i o n t h e o r e m w h i c h later turns out to be v e r y c o n v e n i e n t for the c o n s t r u c t i o n of m o n o i d a l c l o s e d s t r u c t u r e s in c o n c r e t e c a t e g o r i e s , n a m e l y in t o p o l o g i c a l spaces, u n i f o r m spaces, m e r o t o p i c s p a c e s and n e a r n e s s spaces. E n l a r g i n g a t h e o r e m of B o o t h and T i l l o t s o n [2] it is s h o w n t h a t t h e r e are a r b i t r a r y m a n y (non s y m m e t r i c ) m o n o i d a l c l o s e d s t r u c t u r e s in t h e s e c a t e g o r i e s , h e n c e t h e r e is a g r e a t d i f f e r e n c e t o the s y m m e t r i c case, w h e r e c l o s e d s t r u c t u r e s s e e m to be u n i q u e (cp. C i n c u r a [3], I s b e l l [8]). A f u r t h e r a p p l i c a tion of the e x t e n s i o n t h e o r e m is a c r i t e r i o n for m o n o i d a l - resp. c a r t e s i a n c l o s e d n e s s of M a c N e i l l e c o m p l e t i o n s . Of c o u r s e a s y m m e t r i c m o n o i d a l c l o s e d s t r u c t u r e is u n i q u e l y d e t e r m i n e d by its v a l u e s on a f i n a l l y and i n i t i a l l y d e n s e s u b c a t e g o r y , but also the c o n v e r s e statem e n t is true, i.e. m o n o i d a l c l o s e d s t r u c t u r e s can be o b t a i n e d by e x t e n d i n g a s u i t a b l e s t r u c t u r e f r o m a s u b c a t e g o r y to its M a c N e i l l e completion.

O. P R E L I M I N A R I E S Throughout

this

paper

we h a v e T - i n i t i a l i.e.

for

every

(fi

: A +

= Tgi'

said

to c a r r y

that and

there for

gi

an i s o m o r p h i s m way maps

Ai,


in C.

indexed)

that

every

lifted

structure

sources

16OBC

(cp.

so

[13],[7]),

map

y

: TB ÷

TA with

1 : B +

A.

A is

respect

with

a one

to all x.. We a s s u m e i e l e m e n t u n d e r l y i n g set

to be a m n e s t i c

Topological

are T - f i n a l

functor,

is a s o u r c e

to a m o r p h i s m

with

T is s u p p o s e d

iEObC).

there

a topological

TAi)i£ I there

such

object

reasons

implies

are f a i t h f u l ,

(class

can be

initial

is a u n i q u e

technical

of

Set d e n o t e s

(x i : X ÷

Tf i = xi,

: B ÷ the

: C +

liftings

source

Ai)i6i,

TfiY

T

functors

liftings

I is a r e p r e s e n t i n g

of

defined

sinks,

object

( Ti£ObSet

all

for

in this constant

for T, h e n c e

we

101

can

identify

category

elements

a£TA with morphisms

[FCu(K,Y)

(K,Y)

to all FHK(x,IR)

i.e.

struc-

by 1.2(a)

space HK(x,Y) is conHK (f'Y) > Cu(K,Y) to

of all HK(x,Y)

we have

in

products

closed

H K to be right

X,Y the f u n c t i o n

the

Hausdorff

structures

of all finite

a partial

Unif.

on

compact

closed

of M.

yields

that the

constructed

subclasses

K

with

Moreover

f : K ~ X continuous,

empty

[8],

functor

[8], p.46),

by K e the class

source

topology with respect

function

Isbell

is

induced

structure

a monoidal

is the set of all u n i f o r m

It follows

Top.

Hausdorff

class

to C, K e, C,

from K to Y p r o v i d e d

K£Ke,f

(cp.

sources.

(cp.

subclasses

®/(UnifxK e)

respect

by r e q u i r i n g

be initial,

initial

are as m a n y d i f f e r e n t

are n o n - e m p t y

to a m o n o i d a l to ®K,

Z

forgetful

For uniwhich

uniformity

to the

and yields

The

= K~K',

M to be a rigid

Take ~ # KcM and d e n o t e

(®p,Hp)

spaces

to p r e s e r v e

we have K®K'

Take

of spaces

initial

X®Y,

following

spaces.

Proof:

the

is left adjoint

[8], p. 46).

is known

2.5 Theorem: Then

with

spaces.

product

from XxY to spaces

product

on f u n c t i o n (cp.

K,K'

to set up the

Unif

functions

semi-uniform

of u n i f o r m

"semi-uniform"

set XxY p r o v i d e d

convergence

structure

the

= Cco(N,R).

Unif

the c a t e g o r y

X and Y we have

by all s e m i - u n i f o r m p.

from HN(N,R)

with

of 2.4,

monoidal

the

hence closed

corresponding distinct

non

structures

in U n i f . Now

let us have

categ o r y

Near

peat

the r e l e v a n t

pair

(X,~)

fying

the

a look at the c a t e g o r y

of nearness facts

spaces

on these

(cp.

Mero of m e r o t o p i c and at the

for example

categories:

[7]).

A merotopic

such that X is a set and ~ is a set of covers following

conditions:

We shortly space

re-

is a

of X satis-

110

(i) If A£~ AcB, (ii)

#~

(iii) The f

and

A6~,

: (X,~)

, i.e.

for each AEA

there

~ are

(Y,~)

called

uniform

is a set m a p

covers,

a uniform

f : X + Y satisfying

f is f i n a l

iff

the

f-1(A)6~.

For

a set I and m e r o t o p i c

iff

is an B6B

with

A^B = { A D B I A 6 A , B £ B } 6 ~ .

implies

of

+

A

e~ements.

that

non-comparable

3)=>

Because

I)=>

2)

are

direct

obvious.

^

decompositions

D is a W - d i a g r a m

there

Djl

. Djl~

=

the

n

n

morphism ^

D ÷

that

n

^

f:

n

D=

D +

XleDjl

@ S. D. 1=I J1 31

^

@ Eo D. along 1=I J1 Jl

"

@ S. O. ÷ ~ E. O. . M o r e o v e r 1=I J1 J1 1=I J1 J1

the p r o j e c t i v e

^

~ S. O. ~ Ker 1=I J1 31

f

and

^

n

xe

^

obvious

Z D k such k<j I

is a t,omomorphism n

lifts

cover

jl,..,Jn~i

separable.

3) C h o o s e

which

e~ements

x=x1+. " .+x n .

@

S.

i=I

D.

J1

In

.

J1

the

following

sense

the

remaining

first

example

is of

too g r e a t

cardinality. 3.4.

Definition.

the rank of D.

Let

D:

I ÷ F-Mod

be a d i a g r a m .

Then

dim F D O is c a l l e d

137

Now Proof

we are

prepared

of T h e o r e m

by i n d u c t i o n

W.

there

Let are

for

the

(x

I n e ~ } be a g e n e r a t i n g set for D . Then n I ~ o free direct s u m m a n d s p~nj of D such that

D= P ( 1 ) e . . . P ( n ) e D ( n )

and

is a pure

of D the

(D/P)

o

subdiagram

= 0 we must

In a b e l i a n

Pontryag~n's well

for

have

group

.,xnep(1) . . .e quotient

points

..ep(n) diagram

Because is flat.

P= @ p(n) ne~ Since

D= P.

o

theory

Theorem

crZterZon for f r e e n e s s

downward

branching

xI .

filtered

posets

W is u s u a l l y [4].

For

or for

proved

diagrams

posets

with

by e s t a b l i s h i n g

this

seems

finitely

to work

many

only.

REFERENCES [I] H.

Bass:

Finitistic Trans.

Brune:

On p r o j e c t i v e

[2]

H.

[3]

P.C.

Eklof:

Monthly [4]

L.

Fuchs: 1970,

[5] M.

[6]

dimension

rings.

83

(1976)

Infinite

and

Math.

Whitehead's

Soc.

a generalization 95

(1960)

representations problem

of s e m i - p r i m a r y

466-488

of o r d e r e d

is u n d e c i d a b l e .

sets.

Amer.

to appear

Math.

755-788

Abelian

Groups

I,

II.

Academic

Press,

New

York

1973

H~ppner

and

ordered

sets

-,-

Amer.

H. Lenzing: are

: Diagrams

theory.

over

Abelian

Springer [7] B. M i t c h e l l :

free.

Leet.

J.

pure

ordered

Group Notes

Rings

Projective

with

appl.

sets:

Theory Math.

Alg.

a simple

(Proc.

874

several

diagrams

over

20

(1981)

model

for

Oberwolfach

(1981)

objects.

partially 7-12 abelian

Conf.

group

1981),

417-430 Advances

Math.

8 (1972)

1-161 [8]

: Some Bull.

[9]

Math.

: A remark (1981)

[10]

applications

Amer.

S.

of m o d u l e 84

Shelah:

on p r o j e c t i v e s

Infinite

abelian

Israel

groups,

Mathematik-Informatik

Universit~t-Gesamthochschule-Paderborn D-4790 Germany

to f u n c t o r

categories.

867-885

in f u n c t o r

J. Math.

HUppner

Fachbereich

theory

(1978)

categories.

J.

Alg.

69

and

some

24-31

constructions.

Michael

Soc.

Paderborn

Whitehead's 18

(1974)

problem,

243-256

APPLICATIONS

OF

CATEGORY

THEORY

M. Hu.~ek,

Applications usually

two

application one

must

here

one

the

the

structures,which

use

special

such

the

with

of

some

new

and

ordered

stigation

Mannheim

in

theory

in

the

conglomerate

natural

set

Unif

from on

two

or and

category

all

for

emon

topological

the

end

lower I will

subcategories

Unif

of of

inve-

Top. Topology

held

in

monotransformations, properties

some

arranged

Set. By ~

over

monotransformations

n:F----+1K

be

the "greatest

Categorical epi

spaces)

may

possibilities or

show

or b i r e f l e c t i o n s of

closed

and to

topological

coreflection).At

a topological all

Top

(or

definitions

monomorphisms)

in

ordered

(e.g.that

"lattice"of main

straightforward

(also

suggest

coming

a

the

of

and

Conference

K be

structures. I want

any

have

convenient

be

coreflections

a

its

the

spaces

be

structures

and

not

cartesian

Unif

into

repeat

are

not

special

of

properties

need

intrQduced

me

nX

uniform

STRUCTURES

need

spaces

all

concerning

Top. Let

and

components the

ago

I have

Unif the

additional

years

for

of

of

all

some

background

coreflections:

some

category

[Hua].Let

uniform

Unif

functors

see

note

in

conglomerate

results

to

properties

and

coreflections

add

Six

procedure

reflections

category

bound

theory

UNIFORM

Praha

a categorical

in

into

spaces)

category

often

concerning bedded

of

components:

TO

we

de-

(i.e.all

the

F,together

with

order nIS, T~S.

if if

~T)

.

Prox w i t h

category

composed

of

all

totally

the

full

bounded

spaces.

Let (Z,_ (ii)

(e'',m)

such

m(tm)

I ~e : (e,m)

EN M

for

[-facto-

only.

is closed under composition.

E

6 E and

also

from

and m

but

1.2:

(ii)

Proof:

t

. Thus

, then

rization}

let

,

rigid,

i

m

Under the assumption are equivalent:

Proposition

(i)

I

i

e''

(tm) e'' = tf = e'' it

~]

m

is

m t : e'

show

that

orthogonal

by

the

composite

. By

1.1,

mC

of

Lemma E

e'e e'e

1.1

. By

(i)

exists,

. There

one

has

, m

is

an

.

E-factorization Since,

that

E-factorization

Lemma E E

are orthogonal.

the

locally

it

(e',m')

(ii),

orthogonal

suffices

e'e 6 E

to

of

m

one

has

show

[-facto-

that,

for

, necessarily a unique

e' t

that e'e

I

I

t

m'

I

e commutes. zations

From

easily

gets

is w e l l

known

Remark:

The

equivalence

for

factorizations

7.3,

now

rigidness

(iii) = > (i)

Lemma

one

the

~ of

the

the

two

locally

equations

(and

of

m

of

te ' = I

trivial).

(ii),

(iii)

cones.

orthogonal and

E-factori-

e't = I

[]

has

But

in

been that

proved case

in the

[13] proof

a

178

is e a s i e r

since

then one has

phisms whereas,

We mention tiple for

to c o n s i d e r

in the p r e s e n t

without

pushouts,

if

proof

K

its m o r p h i s m s .

that

admits

More

E

E

E

of e p i m o r -

is a r b i t r a r y .

is c l o s e d

locally

precisely,

f

only classes

context,

under

orthogonal

pushouts

and mul-

E-factorizations

if in the p u s h o u t

diagrams

A

e

le I ip

e °

.

f'

B.

~C

1

Pi we have

e 6 E

and

e. E E

f o r all

i6 I

(I a n y

set)

, we

also have

1

e' 6 E the

and

d6 E . Vice

existence

phisms.

of

This was

[13 ], C o r o l l a r y We

conclude

orthogonal stence

locally

Consider

this

pushouts

E-factorizations

generally

section

adjoints

K , and

a commutative

of t h o s e

for

~/ields

of all m o r -

factorization

of c o n e s

in

6.5(I) . by

showing

that

the

can be equivalently

to v e r y

the c a t e g o r y

of

the e x i s t e n c e

orthogonal

shown more

E-factorizations

of r i g h t

morphisms

versa,

MorK

natural

of

by the exi-

locallY

functors.


a morphism

existence described

of

[u,v] : f ~ g

K in

: objects MorK

are

the

is g i v e n b y

K-diagram r f

Composition MOrEK

whose

is t a k e n objects

g

from

K . MorK

contains

are a l l m o r p h i s m s

in

(ii)

MOrEK

is coreflective in

M o r K.

MOrEK

is coreflective in

Mor K

subcategory

with coreflections

[1,m].

Proof: O n e e a s i l y p r o v e s t h a t E-factorization gives

full

1.3:

(iii)

of type

the

E

The following assertions are equivalent: admits locally orthogonal E-factorizations for all

Proposition (i) K morphisms.

*i

i

of

(i) < = > (iii).

be any coreflection.

(e,m)

f , iff

[1,m] : e - f

In o r d e r

to s h o w

Since

1 6 E

is a l o c a l l y

orthogonal

is a c o r e f l e c t i o n .

(ii) = > (iii),

the morphism

let

This

[k,m] : e ~ f

[1,f]: I ~ f

factorizes

179

as

[k,m][r,s]

Now, t h e

= [1,f]

with

[1,e],[rk,sk] are

equalized

particular chosen

e

by rk=

as a n

Consider are

all

the

kr =I.

: I - e

coreflection

I. T h e r e f o r e ,

identity.

the

(e,m)

a morphism

commutative

[ 1,m]

k

is an

, so t h e y

must

isomorphism

and

be

equal;

may

be

in

even

[]

category

pairs

, and

jr,s] : I - e ; in p a r t i c u l a r

MOrEK-morphisms

two

Fact

such

K

of

that

[r,t,s]

the

factorizations domain

: (e,m) - (p,n)

of

m

of

K

is t h e

in F a c t

K

: objects codomain

is g i v e n

of by

a

K-diagram r

el

t

ip j,.

mI

In .L.

Composition all

is

taken

from

E-factorizations

and

FactEK-MorK

:

K

(i)

. Let

FactEK

be

the

full

subcategory

of

let

, (e,m)~me

VE b e t h e composition

PropoSition

K

£

functor. The following assertions are equivalent:

1.4:

admits locally orthogonal

E-factorizations for all mor-

phisms. (ii)

VE

has a right adjoint.

(iii)

VE

has a full and faithful right adjoint.

Proof: T h e e a s y p r o o f torization

of

f

counit

VE

at

of

immediately (ii)

~

the

(iii)

: VE(e,m)

(1,f)

-

(e,m)

hs =

satisfy equal.

I.

the We

that

f)

to

is

left

consider

- f. with The

have

rk = []

is a l o c a l l y

There

to t h e

of

(i)

an arbitrary

=

and

sh =

= I,

orthogonal

From

this

So we

one

still

VE-Couniversal

[1,1],

[rk,tk,sh]

(i.e.

a

obtains have

to prove

morphism [r,t,s]


[k,h]~TE[rk,tk,sh], so t h e

f-fac-

VE-Couniversal

FactEK-morphism

[I,e,I],

[k,h]VE[1,e,1] I

(iii).

is a u n i q u e

[k,h]VE[r,t,s]

is

reader.

and

FactEK-morphisms

equation

is a n i s o m o r p h s m .

(e,m)

[1,1] : V E ( e , m ) - f

equivalence

and

[k,h]

and

iff

kr

: =

I

: (1,me) - (e,m) so t h e y

VE-Couniversal

are

morphism

180

The

third

characterization

E-factorizations pushouts (e,f)

by a d j o i n t

in K. Let

with

is g i v e n

common

RelEK

be

domain

the

and

by a c o m m u t a t i v e

of the e x i s t e n c e

functors

requires

following

e c E, and

of

locally

orthogonal

the e x i s t e n c e

category:

of

objects

a morphism

are

pairs

{u,r,t}: (e,f)~(p,g)

K-diagram

u

f

e

Composition

is

PE here

f(e)

'f

r

taken

denotes

~

K. T h e r e

Fact

the

1.5:

(i) K admits morphisms.

and

from

: RelEK

proposition

P

K,

(fixed)

is

a canonical

funetor

(e,f)~(f,f(e));

pushout

of

e along

f.

The following assertions are equivalent: locally orthogonal E-factorizations for all

(ii)

PE

has a right adjoint.

(iii)

PE

has a right adjoint with counits of type {1,1,h}.

Proof:

(i) ----~ (iii)

Let

(e,m)

be a l o c a l l y

orthogonal

let

the p u s h o u t

property

~

there

(f,u)

be

an o b j e c t

in F a c t

E-factorization

is a u n i q u e

h

such

that

turns

out

of

K

uf

. By

" < ~ - ~ _ f(e)

•

commutes.

Now [ 1 , 1 , h ]

: PE(e,f)

~

(f,u)

to

be

PE - c o u n i -

versal. (iii) one

can e a s i l y

torization

versal. Fact

~

(i) G i v e n

show

of

f

(ii)

~

that

[I,1,u]=

[j,k,h]

In p a r t i c u l a r

(iii)

and

tk=1

Their

PE-images

PE

Let

gonal

orthogonal

[j,k,h]:PE(e,g) that

{u,r,t}

one

j

and

looks

and at the

the d u a l

a class

M-factoriation

~

M

(e,m)

kt=1 two

: (1,g) by

(f,u)

are

with

u=1

E-fac-

be P E - c o u n i -

isomorphisms.

(f,u)

The

can be w r i t t e n

{u,r,t}: (1, f) ~ To p r o v e

the

as

(e,g)

equations

RelEK-morphisms --

(e,g)

[j,k,h]

concept

~

k

by a u n i q u e

jr=1

are e q u a l i z e d

We m e n t i o n For

is a l o c a l l y

(e,hf(e))

{e,1,1},{erj,rj,tk}

torizations:

arrow

[1,1,u]:PE(1,f)=(f,1)

we h a v e

rj=1

a PE-Couniversal

.

It suffices to s h o w

K-morphism

such

of

, so they locally

of m o r p h i s m s is a l o c a l l y

of

are equal.[]

orthogonal

E-fac-

K , a locally coortho-

orthogonal

M°P-factori

-

181

(m,e)

zation

property

in

K Op,

pictured

that

is, one has m C M

and the d i a g o n a l i z a t i o n

by

e

m

'T

"i

It

g

h

I i v

I

n6M 2. E x a m p l e s

We r e s t r i c t t oriz a t i o n s general:

which

ourselves

are not o r t h o g o n a l

the c o n s i d e r e d

f

and

for all

x

regular,

iff every

zation ker

and

g

(e,m)

f ~ker

with

common

y , fx= fy

of

g

e. Now,

for

E-fac-

The first one is

for all m o r p h i s m s

E

(Isbell[6],

Kelly[7]).

domain we w r i t e

implies

with

gx =gy

of any

iff

e

the class

ker

For two

f £ ker g , if,

. An e p i m o r p h i s m

ker e ~ ker g factors

is regular,

f

exist

orthogonal

and coequalizers.

Regular factorizations

2.1

of locally

a priori.

factorizations

category with kernelpairs

morphisms

to e x a m p l e s

as

g=he

is a regular

of all regular

e

is

. A factori-

epimorphism

with

epimorphisms,one

is a locally orthogonal E-factorization (hence unique up to an isomophism) and that a locally orthogonal E-facotrization is regular, if K has coequalizer~.Furthermore, if K admits regular factorizations, the following assertions are equivalent: easily

shows

that every regular factorization

In every regular factorization (e,m), m is a monomorphism. (ii) Regular epimorphisms are closed under composition. (iii) K admits orthogonal E-factorizations for all morphisms. (i)

The c a t e g o r y these

are

regular

Cat of small

locally

orthogonal

epimorphisms

the dual

of the c a t e g o r y

many

factorizations

there

is no o b j e c t i o n

zations. fibres.

to obtain

So let Then

continuous iff there factorize

E

mapping is a f

y C Y

factorizations;

since

in Cat.

The

compositions same holds

of

in

of semigroups.

have

their

to c o n s i d e r i n g orthogonal

but

be the class

Y

just

range

orthogonal

factori-

in Top with

E-factorization

connected for every

Xo,X I in X equivalent,

C in f-ly with Since

But

as long as one does

maps

: call two points

map.

[4]

of m o n o t o n e -

to all T1-spaces.

locally

orthogonal

and a c o m p o n e n t

over the q u o t i e n t

from E i l e n b e r g %

all spaces

of q u o t i e n t

a locally

f:X ~

Arising

found g e n e r a l i z a t i o n s

and e x t e n d e d

Top admits

regular

but not o r t h o g o n a l

factorizations.

authors

light

not e x p e c t

admits

need not be r e g u l a r

2.2 Monotone-light investigations

categories

Xo,X I C C, and

E is not closed under

com-

182

position,

this

analogous

results

factorization

connectedness.

are

That

is not

obtained

in this

case

may

be

x6R

,x~O}u{(O,y) :-I~y~I}~R 2

seen

singleton

from

space

In a m o r e

the

first

E

onto

complete

context

concrete

of

K , that

tains

a non-void

one

of

K

is

A

object

For m o r e to B ~ r g e r ' s

and w h i c h

considerations

[2].

are p o s s i b l e

such

admits

that

each

locally

~x,sin!) :

map

onto

a

fibre

2.3

We

of

of

K be

K

under

[5]

the

also

for

but

Let

an initisubcate-

which

epimorphic

K

Then

conimages

admits locally

is an i m m e d i a t e

conse-

1.2 above. interested

mention

E

the

is c o n t a i n e d

orthogonal

any d i f f i c u l t y

the class of all quotient maps

MA

for S t r e c k e r ' s

factorizations; for i n s t a n c e , maps

of

composition

be a c o n n e c t i o n

subcategory

investigations

thesis

A

is c l o s e d

A . Theorem

general

without

of s u b o b j e c t s .

and of P r o p o s i t i o n

recent

let

is a full

MA-factorization for

of this

succeeding

can p r o v e

and

families

with fibres in

quence

and the

under

subspace

for t e r m i n o l o g y ) :

category

of c e n t e r e d

orthogonal

~

closed

of the

Completely instead

. general

gory

and u n i o n s

p

in g e n e r a l .

connectedness

is not

projection

(cf. H e r r l i c h - S a l i c r u p - V ~ z q u e z [ 5 ] ally

orthogonal

for p a t h w i s e

that

[11] class

is r e f e r r e d

corresponding

submonotone-superlight of t h o s e

in a p a t h

not n e c e s s a r i l y

reader

quotient

component,

orthogonal

Top

E-factori-

zations. 2.3 Eilenberg-Moore factorization. and

functors

right We

we

adjoint

claim

consider

functors

that

the

and

(non-full)

its

subclass

In a c a t e g o r y subcategory M

Cat of c a t e g o r i e s

Catra

of all

of all m o n a d i c

functors.

, every right adjoint functor U:A ~X has a M-factorization, provided A and the Eilenberg-

in C a t r a

locally coorthogonal

Moore category x T ( T is a m o n a d i n d u c e d by U) have coequalizers. Proof: It is k n o w n that the e x i s t e n c e of c o e q u a l i z e r s in A makes

the

if we

consider

comparison the

functor (outer)

• XT

!

S

~

XT

have

a left

adjoint.

Now,

diagram

UT

K A

P

K:A

commutative

~X

1

'

IV I

Id

~ ,yS

US

l

R

~y

in C a t r a for any m o n a d with VK= P and uSv=

$=(s,~S,~S), then t h e r e is a u n i q u e f u n c t o r V RuT; we r e s t r i c t o u r s e l v e s to s h o w u n i q u e n e s s :

for

V(X,x)

a T-algebra

structure

y:

(X,x),

SRT -- RX,

m u s t be w r i t t e n

and the T - h o m o m o r p h i s m

as

(RX,y)

x: (TX,~X)

with ~

an S-

(X,x)

is

183

tralnsformed

into

V(TX,~X)

an

= VKFX

S-homomorphism = PFX

=

Rx:

(uSpFx

V(TX,vX)

-

(RX,y).

Since

, u$cSpFx)

t

(where

F

is

left

adjoint

y = y • SRx • SRqX (One R

notices

to

show

proof.)

a

One an

does

uniqueness has

arguments

one

not of

use V;

the

it

for

right

is,

coequalizers

(apply,

necessarily

obtains

• SRnX. adjointness

however,

V

has

instance,

used

a left

of

in

Corollary

7 of

P

the

adjoint

New factorizations from old by adjoint functors.

functor

let of

one

XT

U)

and

existence

by

adjoint

[12]

to

Let

U:

= RUT).

2.4 be

the

Since

triangle uSv

that

to

= Rx • uSeSPFX

E' all

with

be

the

left

pushout

morphisms

then

in

the the

F.

For

of

FE

pushouts

of

closure

which

obtains

analogue

adjoint

are

following

orthogonal

lifting

a class in

E

A,

of

that

a morphism theorem

is, Fe

which

A -

X-morphisms, E'

consists

with does

e E E.

not

have

case:

Let A have pushouts and let E consists of X-epimorphisms only. Then, if X admits locally orthogonal E-factor~zations, A admits locally orthogonal E°-factorizations. Theorem

2.5:

Proof:

In

order

f:

A -- B

we

form

Uf

and

then

the

to

find

the

an

locally

pushout

E'-factorization orthogonal

is

claim

that

So w e

consider

a unique

m' : C - B

(e'm')

is

with

a locally

a commutative

FUf.

upper

Fx

y: =

f

and

m'h

=

eB • Fm.

E'-factorization

of

We f.

E

b

A

X - UA,

=

Iv .,

a

eB.

m'e'

orthogonal

~ FY

D

x:

of

diagram

Fp

ul the

(e,m)

C

FX

where

A-morphism

FZ

A

There

the

diagram

Fe

FUA

of

E-factorization

----~C part

Y - UB fau

m is

~ B

a pushout

with

= bv.

Fp

eA=

eB.

with Fx

= au,

Fy.

Fp

p 6 E. W e eB • F y one

have = bv.

derives

morphisms From fx = y p .

Hence

X

184

there

is a

perty,

an

t: Y - Z

of e p i m o r p h i s m s , bp'

one

gets

Remark:

also

U

finite number

2.5 is v a l i d

This w a s

To this

limit n u m b e r

~ Sl

than

(el,C)

, if

composite

(el,C)

exists

(and

is said to be in

to a

E

if

(of a m o r p h i s m

f = m e l)

(el,C)

(el,C)

l-factorization

E . A

f)

there

is a u n i q u e letting

either

B = e +I

B

D(O1)

•

~.ee

. , , . , .

,

C(O1 )

and

mt~ =h

(E,l)-factorization

((el,C),m)

The

con-

such that

me l if in

is orthogonal diagram

•

•

.

,.

6

,

as p i c t u r e d

,

I

~I

m

transformation

is c a l l e d

hi

D E

(pl,D) ± ((el,C),1~J

and

by the d i a g r a m

tl, ,'' C(~l)

natural

mt I = h

O ~ ~< 8S I

D(yl)

~.

. We w r i t e

colimits.

such that

whenever

. ~It . .B. . . . . .t~ ,;'

is a u n i q u e

(of the

~i h

,...,t I

a limit ordinal,

,

of m o r -

((el,C),m)

m

(pl,D)

m

C(e6)

T h i S m e a n s : ther e t °o = g

'

tl,t2,..,t

t ~I~,

as for

is a f u n c t o r w i t h

if for each c o m m u t a t i v e

D(~B)

g ~. . . . . . . . t .l . .',.

set

([,l)-factorization,

, tBD(aB) = C ( a B ) t a

or

6

I

eI

sequence

to =g

to

l-composite

is a pair

Pl

and,

number

e a8 @ [ (e < l;B = ~ +I)

l-composite

((el,C),m),

gi

~

the a b o v e

and a m o r p h i s m

. It is c a l l e d an

is in

from

C : I +I ~ K

an in-

I , considered

is the

which preserves

addition


the o r d i n a l

less

(and

of a c o m p o s i t e

factoriza-

end we i d e n t i f y

e~8(e < I;8 = ~ + I ) )

l-factorization

=

(in the d u a l si-

into p o s s i b l y

C(Ol) = e I

A

shown

is the c o l i m i t of the

. A pair

C(e8) = e eB)

m'sp'

of a m o r p h i s m

morphisms

sisting

so from

for r e g u l a r

first

be the u n i q u e m o r p h i s m

y6(y < K;6 = y +I)

consists

2.2.

set of o r d i n a l s

~6

E

[]

([,l)-factorizations

of factors.

Since

of e p i m o r p h i s m s ;

factorizations

and let

sv = h • Ft.

is u n i q ue.

Lemma

orthogonal

w i t h the o r d e r e d

~ 6 . Each

s

of T h e o r e m

[10],

W e next c o n s i d e r

phims

consists

is f a i t h f u l .

by M i t c h e l l

a category,

sp' = e'a,

E'

The a n a l o g u e

3. L o c a l l y

I~ I

tp = ex, m t = y, and, by the p u s h o u t p r o -

with

m ' s = b', and

tions provided tuation)

with

s: E -- C

t :D~ C

with

in this case.

loeclly orthogonal

, if

An

185

(pl,D) ± ((el,C),m)

for all

(E,1)-factorizations E-factorizations Analogously thogonal has

whenever

canonically t : C-C'

with

and

We did not i n t r o d u c e reason:

respect

Proposition

C(O1)

= e

Proof: in

and

unique

E =

1.2,

h ~ I

•

.°

.

.

.

,

m

C:

B = a + I

(e,m)

I + I ~ K

that

(Ph ' D)

be d e f i n e d

• ((e,C),m)

whenever

the

±

•

,

°

.

.

.

.

I I t8

for all

hph = meg,

•

•

P

following p

•

diagram

t

=

tBp

~8

is a

•

i tl I v

commutative I 1

Ihpl m

for

m

O < ~ < B ~ l

The o r t h o g o n a l i t y

pOl C E. Thus

~tl I

i

&

I

eg I

(pl , D)

there

I

since

oi

by

B~

I t61

.&

(1,m)

for

such that in the d i a g r a m

8 = limit ordinal.

pOl

may be r e -

((e,C),m)

I ~ a ~ 6 ~ ~.

, mt I = h, and

or

with

composition.

E-~ctorization

J

oi

Therefore,

(E,l)-factorizations

under

I

eg = t l P

are

equivalence

(E,l)-factorizations,

(E,l)-factorization

I t~l

+

(E,I)-

that e v e r y l o c a l l y o r t h o g o -

p~B =

natural

I

s h o w that,

e

rendering

first

is not c l o s e d

for

show that

I iltl

implies

of o r t h o g o n a l

by l e t t i n g

oi

g

and

(unique)

locally orthogonal

tl,t2,...,t h

I

we h a v e

locally orthogonal

(E,1)-factorizations.

£

= I

is, we m u s t

sequence P

.

orthogonal

C(a6)

We m u s t

E. T h a t

m'tl =m

3.1: An o r t h o g o n a l

any f i x e d o r d i n a l

is, one n e c e s s a r i l y in the a b o v e d i a g r a m .

I

o n l y if

g a r d e d as a (locally)

l o c a l l y or-

is of the f o r m

orthogonal

to P r o p o s i t i o n

that

of the same m o r p h i s m

is a

I

m a y be i n t e r e s t i n g

(locally c r t h o c o n a i )

that

h =m

there

We w i l l p r o v e

e admits

, and

a concept

(E,l)-factorization

K

are rigid,

((ei,C'),m')

i.e.

to = I

the

(locally orthogonall

one e a s i l y p r o v e s

, g =I

(ek,C),m),

isomorphic,

for a s i m p l e

with

of this one has that two

factorizations

if

I =1

D =C

E . The

I

to the case

As a c o n s e q u e n c e

in

m a y be i d e n t i f i e d

of S e c t i o n

(E,k)-factorizations

t =I

nal

(pl,D)

there

of

(e,m)

is a u n i q u e

tI

186

6

Let for

be

an

I < a < when

t 8 =

ordinal 6

~ =

a + I

diagram

we

have

6 + I

then

with

I < 6a

top

or when

there

is

6 ~

~

and

= t6

and

is

a limit

a

t8

a unique

suppose

inductively

= h p al

mt

ordinal

rendering

for

and the

that

a unique

1%

6 < ~.

If

following

commutative pa6

I

l

It 6

h P 6~

1

since

pa6

ty

K,

in

But

in

•

(1,m).

there

the

is

If

addition

6

> . For a normal monomorphism and a normal epimorphism we

shall use the symbols ~

> and

~, respectively. If

exact sequence, then the factor object

C

B ~

will be denoted by

> A

~ C

is an

A/B.

2. Axioms on categories Several systems of axioms have been imposed on categories in order to enable the development of a general radical theory. Many of these systems of axioms aim at obtaining categories similar to those of rings or

~-groups - similarity means

that one can use techniques, in particular the iscmorphism theorems in their familiar forms, like in the latter categories ([i] , [2], [3], [5], [6], [ii], [12] , [13] , etc). Some other authors, on the other hand, tried to develop general radical theory in a more general setting; accordingly they could not get too far without making further restrictions either on the category or on the radicals under consideration (for recent investigations in this direction we refer to [14] ). Here we want to give a clear picture on the present situation in ring-like categories. Almost all the systems of axicms used in them for the purposes of radical theory, are slighter modifications of that of ~ul'ge~fer [ ii] . We shall use the form presented by Andrunakievi~ and Rjabuhin [2] : it is supposed that the considered category

C

satisfies

191

(~a_) (~2) (~)

C

has zero object,

every object possesses a representative set for its normal subobjects, for each object its normal subobjects form a c(mplete sublattice under the natural partial order of subobjects,

(AR4)

every morphism has a kernel,

(~5) (~6)

evel-y morphism has a normal epimorphic image, normal epimorphisms carry normal subobjects into normal subobjects.

We reni~rk that (AR2) is i~i0osed in [ 2] in the stronger form that all subobjects of any object form a partially ordered set, but the present weaker form suffices both for the investigations in [2] and for our purposes. HoloDmbe and Walker [ 5] used a seemingly different system of axioms. Here we exhibit, however, that the two systems are equivalent. First of all, let us have a closer look at the categories considered by Fblcombe and Walker. Part of their theory makes use of the second isomorphism theorem without a clear formulation in the categories for the latter to hold. They considered a category

C

satisfying

the following axiem~: (HWI) (HW2)

C in

has a zero object, kernels and cokernels, C

intersections of arbitrary sets of normal subobjects of an

object exist, further, every object possesses a representative set for its normal subobjects. The intersection of normal subobjects is again normal ([5] Ler~m% 1.5). (HW3)

Every morphism of

C

factors through a cokernel followed by a mono-

morphism. Under these axioms

C

has inverse images of normal subobjects which are again

normal ([5] Lemma 1.4) and has epimorphic images ([5] Lemma 1.9). (HW4)

For every cokernel t (t-i (Z))=Z.

t: X---->Y and kernel

Z --> Y

we have

The first isomorphism theorem is proved to be a consequence of these axicms ([ 5] Theorem i,ii). Finally we ass~ne (HW5)

Any set of normal subobjects of any object possesses a union which is again a normal subobject.

Let us notice that in the presence of the systems of axicms ~WI)-(HW5 ) the normal subobjects of any object form a complete lattice under

n

and

u

For short, the systems of axic~s (ARI)-(AR6) and (HWI)-(HW5) will be denoted by (AR) and (HW),respectively. THEOREM i. (}~) implies (AR), and if any two s.ubobjects of objects in have an intersection, then the cenvers~e also holds.

C

Proof: In view of what has been said above, it is clear that the system (HW) implies (ARI)-(AR5). In order to prove (AR6), let kernel

ker t: Z -->X , and

N

t: X --->Y be a cokernel with

be a normal subobject of

X . By [5] Lenma i.i0

192

t(NU Z) is a normal subobject of u n i o n w e have object of

t(N).

plying that

t(Z)= OS_t(N) , by the definition of t(NUZ)

is a normal sub-

Conversely, by the definition of the image

t(N)=t(NU Z)

is a normal subobject of

Conversely, notice that case of (AR6)

Y . Since

t(N OZ)_~B

through which each

f(A)NB' exists, it contains all the

f~

1

f(A ),

1

containing each A i , f' A ~ f(~> > B , and

f(A)

by

C

be

f(Ai) . Suppose that

factorizes. Then, provided that and since the

f(A ) are normal

1

subobjects of

iE I ,

1

(~R6), they are normal in

f(A)NB',

too, whence

C_C ,

and conversely,

(f')-I(c)>A. 1

for all

iEI

conclude that izes through

, hence f(A')

(f')-I(c)>A' and

C

,

thus

C=f'(f')-l(c)>f'(A')=f(A ') . So we

are equivalent subobjects, whence

fe

also factor-

~ , and we are done.

Remark i. By [ 2] p. 397 we know that the second isomorphism theorem holds under (AR), thus it holds under (HWI)-(HWS) as well. Remark 2. Krer~oa and Terlikowska [ 6] and then Terlikowska-Os~owska [ 12] , [ 13] introduced a self-dual system of axioms which is satisfied in the categories of associative or alternative rings but not in that of not necessarily associative ones. Hence this system of axicms cannot be equivalent to (HW) or (AR), nevertheless there is a strong connection between them. Stlopose we are given a category C

satisfying the system of axic~s (AR). Consider the subcategory

of all objects of cc~positions.

C

C'

consisting

and those morphisms which are kernels or cekernels or their

(For establishing the basic results in the general radical theory

of rings, only these morphisms are really needed. ) Now it is straightforward to

193

check that

C'

satisfies the system of axic~s

AI-A6,

A6 ¢~ and A7 ~'~but not A7

of Terlikowska-Os~owsk~ [12] . Conversely, AI-A7, A6 ~, A7 ~'~of [12] inply most of (AR) but (AR4) and (AR5) only in a weaker form. In the rest of the paper we shall always work in a category

~

C

satisfying

(~). 3. Radicals In his paper [ 3] Carreau presented an elegant treatment of radicals in cate-

gories. What he did in genuine categorical terms is, expressed in the classical language, that radicals can be defined beth by means of a function and of a semisimple class. The same idea is basic in Hoehnke's earlier development [ 4] of radicals in categories of universal algebras. In the category of associative rings Michler [8]

introduced a notion of radical at the same time as Hoehnke did for

universal algebras; these two notions are equivalent for rings. Now we present Carreau's definition of a radical functor in the slightly modified but equivalent version given by Holcombe and Walker [ 5] . By the cokernel subcategory jects are the objects of

C

E(C)

of

C

we mean the subcategory whose ob-

and whose only morphisms are the cokernels of

C .

(In Carreau's terminology t h i s is a special coextensive subcategory. ) A covariant p: E(C)--->C

functor (i)

p

(ii)

is called a radical functor, if

for all

CE C ,

(iii) p(C/~(c))=o

p(C)

is a normal subobject in

for all

C6C

normal subobject

p(C)

in

C

~(p(C))C

is a subfunctor of the inclusion functor

p

assigning to each

CE C

a

satisfying (iii) and for any cokernel

~

frcrn C .

Theorem 2.2 of Carreau [3] states exactly that every radical functor defines a radical and conversely. The most ini0ortant radical functors are cc~plete (which means that if p(B)=B

for some normal subobject

(p(p(A))=pB

p(B')#B'

_p(B)/(p(A)AB)=BI(p(A)nB)=B''

,

195

whence

p(B'')=B''

, a contradiction.

The converse iaplication is obvious.

PROPOSITION 5. If a radical functor radical satisfies

p

is idenlootent , then the corresponding

(M2).

The assertion is obvious. Moreover, under the validity of (~iI) the converse iaplication is also true. THEOPd~4 6. Every cc~plete and idemlootent radical functor defines an

4-

radical and conversely. Proof: In view of Propositions 4 and 5 all we have to prove is that the radical functor

p

defined by an 4 - r a d i c a l

idempotent. Now there is an exists a

B/p(p(A))

= B/p(p(A))

AEC

such that

. By condition

B/O(p(A)) 4 p(A)/p(p(A))

is ideml0otent.

such that B/p(p(A))4

p(A)/p(p(A))

(iii), however, we have condition

St~opose that

p(A)/p(p(A))#O

(M_l) yields

and

p

is not

. By (M2) there p(B/p(p(A))) =

p(p(A)/p(p(A)))=O

p(B/o(p(A)))#B/p(p(A))

and so for ~ a contra-

diction. As usual, to a radical IR

=

S

= [AEC

P

{A6C

: p(A)=

p

we assign two classes A}

and P

: p(A) = 0},

called the radical class and the semisir~le class of

p , respectively.

Knowing the

equivalence of the previous definitions of radicals, the connection between radical and semisir~ole classes as described in Andrunakievi~ and Rjabuhin [ 2] , V, §2, Theorem 3, yields exactly Theorem 3.10 of Holcombe and Walker [5] and its converse. (The latter is the same as [5] Theorem 3.11; in fact the sufficient condition give~l in the note after this theorem, is always satisfied in view of (AR6). ) Thus an

4-radical

p

on any object

A

can be determined both from below and from

above: u (B 4 A

: B 6 IRp) = p(A) = e (C 4 A or C = 0 : A / C E S p )

Till now we have characterized an

.

4 - radical by means of the radical assign-

merit (radical functor) and the radical class. It can also be characterized in terms of the semisimple class and by the pair of radical and semisimple classes, respectively. Such characterizations

for not necessarily associative rings or

~-groups

exist in plenty (see e.g. [ 7] and [ i0] ), and using the tools we already have in our category, their proofs can be carried out word by word in our case, too. Here we pick out just one characterization of each of the latter two types. THEORI!M 7 class of an

(~]itz [i0] Theorem 4 ). A class

4-radical

if and only if

S

S

of objects is the sen/simple

satisfies the following three conditions:

196

(a) i_ff B 4 A 6 $ , then (b)

S

B

has a non-zero factor object in

(c) for all

AEC

, ((A)S)S = (A)S

where

(A)S=N ( B 4 A

THEORI~4 8 (Mlitz [iO] Theorem 2). The classes and semisimple classes of an I~ ~, S

consists of zero objects,

(B)

A E I~

and

(C)

A E S

(D) for any

and

A/B # O S ~ A

A E C

I~

and

or S

B = O : A/B E S). are the radical

4-radical if and only if

(A)

and

$ ,

is closed under subdirect products,

inloly A/B ~ $ ,

i~ply

B~I~

,

there is a normal subobject

B

o_ff A

such that

B 6 I~

A/B ~ S .

References [ i] S. A. AMITSUR, A general theory of radicals, II, Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100-125. [ 2] V. A. ANDRUNAKIEVI~ and Ju. M. RJABUHIN, Radicals of algebras and structure theory (Russian), Nauka, Mosoow, 1979. [3] F. CARREAU, Sous-cat~gories r~flexives et la th~orie g~n~rale des radicaux, Fund. Math. 71 (1971), 223-242. [4] H.-J. HOEHNKE, Radikale in allg~meinen Algebren, Math. Nachr. 32 (1966), 347383. [5] M. HOLCOMBE and R. WALKER, Radicals in categories, Proc. Edinburgh

Math. Soc.

21 (1978), 111-128. [6] J. ~ A

and B. TERLIKOWSKA, Theory of radicals in self-dual categories,

Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 22 (1974), 367-373. [7] L. C. A. van ~

and R. WIEGANDT, Radicals, semisimple classes and torsion

theories, ~ t a Math. Acad. Sci. Hungar. 36 (1980), 37-47. [8] G. MICHLER, Radikale und Sockel, Math. Ann. 167 (1966), 1-48. [9] B. MITCH~.T., Theory of categories, Academic Press, 1965. [iO] R. MLITZ, Radicals and semisi~ple classes of

~-groups, Proc. Edinburgh Math.

Soc. 23 (1980), 37-41. [ii] E. G. ~UL'GE~ER, General theory of radicals in categories (Russian), Mat. Sb. 51 (1960), 487-500. [ 12] B. T E R L I K O W S K A ~ S K A ,

Category with self-dual set of axic~s, Bull. ~ a d .

Polon. Sci. S~r. Sci. Math. Astronom. Phys. 25 (1977), 1207-1214. [ 13] B. TERLIKOWSKA-OS~OWSKA, Radical and semisimple classes of objects in categories with a self-dual set of axioms, Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 26 (1978), 7-13. [ 14] S. VELDSMAN, A general radical theory in categories, Ph. D. Thesis, University of Port Elizabeth, S. A., 1980.

ON THE STRUCTURE OF FACTORIZATION STRUCTURES by A. Melton and G. E. Strecker

For any category on

K.

K

In particular,

we investigate the family of all factorization structures

for each such structure,

lattice of all factorization structures on

K

(E,M), we investigate the complete with left factor a subclass of

E;

this investigation is based on a Galois connection between all such structures and the lattice of all full isomorphism-closed subcategories of

K.

families are precisely all the E-reflective subcategories of

The Galois-closed

K

and all the (E,M)-

dispersed factorization structures of Herrlich, Salicrup and Vazquez.

AMS

(1980) subject classifications:

Secondary:

§0

Primary 18A20, 18A32, 18A40;

06A15, 18A22

Introduction The importance of factorization structures on categories is by now well

appreciated.

Over the years the conditions that have been considered necessary for

an "(E,M)-factorization structure" to carry that name have evolved from those requiring

E

and

M

to be sufficiently nice dual-like classes of epimorphisms

and monomorphisms such that each single morphism has an essentially unique factorization,

(E,M)-

through various stages until the current generally accepted criteria

that (among other things)

E

be a class of morphisms and

sources such that each class-indexed source has an (E,M)-factorization,

M

be a conglomerate of

(even empty or proper class indexed)

m%d, in the category,

(E,M)-diagonalization holds.

To

emphasize that we require diagonalizations as well as factorizations we call such entities "diafactorization structures." The two major references for this paper are

[HSV] and

[Ho], both of which made

significant contributions to the clarification of the nature of

(dia)factorization

structures. In [HSV] Herrlich,

Salicrup and Vazquez introduced a new type of diafactoriza-

tion structure called dispersed and proceeded to show that there is a bijection between all E-reflective subcategories of an (E,M)-category dispersed diafactorization structures on

K.

K

and all

(E,M)-

This was a generalization of the

result that for nice categories such a correspondence exists between the epireflective subcategories of [Sl],

[S2] , [S4]).

K

and all perfect factorizations

(cf. [Hel],[He2],[Na],[Ne],

It also ~aproved and put into the proper context much of the

earlier work on quotient reflective subcategories,

connectedness properties,

corresponding factorizations

[SV2],

(cf. [C], [P], [SVI],

and

[$3]).

In §i, via a modification of the main result of Hoffmann

[Ho] (cf. also Harvey

198

[Ha]), we show that the development classes of a category

K

precisely those classes,

such that

E, for which there exists an

diafactorization structure on problem of

[HSV].

any E

(E,M)

are

(E,M)

is a

This answers the outstanding open

diafactorization structure,

(cf. [HS2] , [T]).

(E,M)-category

E

must be a class of

As a by-product of this theorem we also have, for

K, an internal characterization of all those

for which there exists a

structure

(Th.l.3).

(see [Ne])

The proof of Theorem 1.3 also provides an alternative proof of

the fact that for any epimorphisms

K

M

(Th.l.9).

D

such that

(C,D)

C

contained in

is a dispersed diafactorization

It is interesting to note that such classes are (to within

existence of the colimits) the "standard" classes of E-morphisms introduced in [SI] and investigated further in [$2]. In §2 we describe and investigate a Galois connection that makes precise the nature of the bijection discovered in [HSV]. of an (E,M)-category

K

Namely, the E-reflective subcategorles

and the (E,M)-dispersed diafactorization structures are

precisely the Galois-closed classes and are complete lattices

(in a suitably large

universe)

General Galois

that are anti-isomorphic with each other

(2.6(2)).

results, as well as special properties involved,

are used to investigate in more

detail the structure of the complete lattice

of all diafactorization structures

(C,D)

on

K

with

C

a subclass of

E.

Q

In particular,

partitioned into a family of complete lattices Q

it is shown that

(called levels)

can also be viewed as a union of complete lattices

§i.

Characterization of Diafactorization Structures Definitions and Notation

(i) In all that follows

K

is

(called images) all of which

have a point in common and none of which meets any level non-trivially

i.i

Q

(2.6(i) (i)) and that

will denote a category, and

Mot K, Iso K

(2.6(i) (ii)).

and

Epi

will denote the classes of all morphisms, all isomorphisms and all epimorphisms of

K.

All subcategories will be assumed to be full and isomorphism-

closed. (2) A K__-source with domain

X

empty and possibly proper) domain (3) K

is a pair

i

in

where I

fi

I

is a class

(possibly

is a K-morphism with

X.

is called an (E,M)-cate~or~ and

ture on

(X,(fi)i)

and for each

K

provided that

E

tion with K__-isomorphisms and

(E,M)

is called a diafactorization struc-

is a class of K-morphisms closed under composiM

is a conglomerate of K-sources closed under

composition with isomorphisms such that: (a)

K

has the (E,M)-factorization property;

has a factorization (Z,(mi) I) (b)

K

belongs to

i.e., every K--source

X~Y.

= x--~Z ~-i~Y. l l M, and

has the (E,M)-diagonalization property;

K__-morphisms and

(X,(mi) I)

and

(Z,(hi) I)

where

e

(X, (fi) I)

belongs to

i.e., whenever

e

E

and

are K--sources such that

and

f

are e

199

is in

E,

(X,(mi)i)

is in

and for each

M

there exists a unique m o r p h i s m i

in

d:Z---~X

i

in

I, h.e = m.f, then 1 1 f = de, and for each

such that

I, h. = m.d. l l

(*)

Y--~-~1 Z

I

d''1

l

X -------9-W. m. 1 1 [If only on (4)

(a) is satisfied,

(E,M)

is called a factorization

structure

K° ]

([HSV])

If

(C,D)

on

A

K

of

K K

is an

such that

E;i.e.,c:X---~Y f:X--+A

(E,M)-category,

is called C

is precisely

is in

with

A

C

in

then a diafactorization

(E,M)-dis~ersed

iff

A

c

structure

iff there exists a subcategory

all the A-extendible

is in

E

morphisms

in

and for each K--morphism

there is some K - m o r p h i s m

g:Y---~A

such that

f = gc. (5)

Let

E C

Mor K

(a)

~(E)

then:

will denote the conglomerate

the property e (b)

in

A(E)

E, then

X fi~y = X ~ Z ~Y. • i is an isomorphism.

e

will denote the conglomerate

the property that if square

i

a K__-object e:X--~Z

in

morphism (d)

~

e

(*) c o ~ u t e s ,

for each (c)

of all sources

that if

in

I

is in

1

of all sources

E

then there exists a unique (*) commutes.

(cf.

[S I]

is called an E-injective

E

and each K - m o r p h i s m such that

having

(X,(mi) I)

and if for each

Y

g:Z---~Y

(X,(fi) I)

is a factorization

i

in

d:Z--~X

and

with

having

I

the

such that

[$4])

object iff for each

f:X---~Y, there exists a K--

f = ge.

is the category whose objects

are members of

E

and whose ^

morphisms

(6)

(e)

A0:KE

Let

C

>K

and

E

hOmK_ (e,e)

of

h = gf.

(a)

iso--com~ositive

(b)

left cancellative

(c)

(f,g)

is the functor defined by be subclasses

K_-morphisms for which

belongs to

are pairs

iff

h

w.r.t.

E, then

f

Mor K Then

C

belongs to E

where

~0(f,g) and let

ge = ef. = f.

(cf.

f, g

and

[Ho]). h

be any

is said to be: C

whenever

iff whenever

must belong to

h

{f,g} C

belongs to

C U Iso K; C

and

f

C;

pushout p r o n e iff (i) every K--source pushout

X

(X, (c i) i )

ci > Y.

di ; Z

d (ii) every 2-indexed K--source, out

with each with

d

(X,(k,c)),

ci in

in

C

has a multiple

C; and

with

c

in

C

has a push-

200 X-c

k

~Y

t

I

Z •

(d) a development

c

with

class

(ef.

[Ne] (t))

c

in

C.

iff

(i) C~__ Epi K, (ii) C

is iso-compositive,

(iii) C

(e) an E - s t a n d a r d (i) C

class

(cf.

is a development

(ii) C

1.2 Remark.

and

is pushout prone; IS I],

[S 2] (tt))

iff

class of E-morphisms,

is left cancellative

w.r.t.

and

E.

The following are some w e l l - k n o w n properties

structure

(E,M)

on

K

that we will use in the sequel.

(i)

E

is iso-compositive.

(2)

E

and

M

of any diafactorization

determine

each other;

in fact

M = A(E).

We next obtain an improved version of the main t h e o r e m of Hoffmann that no conditions w h a t s o e v e r

morphisms steps in

E.

are put on the category

Some major steps of the proof,

however,

K

[Ho]

in

or the class of K-

closely

follow analogous

[Ho].

1.3 Characterization For any category

T h e o r e m for Diafactorization K

and any class

E

Structures

of K-morphisms,

the following are

equivalent: (i) There exists a conglomerate

ization (2) E

structure on

is a development

M

of K-sources

for w h i c h

(E,M)

is a diafactor-

K. class.

(3)

(E,A(E))

is a factorization

(4)

(E,~(E))

is a diafactorization

(5) The following hold:

structure on

(a)

E

(b)

A0: ~

K.

structure on

K.

is iso-compositive; ~K

is a topological

functor (%f%).

(t)

In [Ne] Nel d e f i n e d development classes ulation avoids his smallness condition.

(tt)

In [Sl] and [S_] standard classes of epimorphisms are defined more genz erally, without the requirement of the existence of (multiple) pushouts in what corresponds to (6)(c).

(tt+)

A functor

F:A---~X

has a factorization

is called topological

somewhat less generally.

iff each F-source

(X

gi

Our form-

~FAi) I

(X gi y FA.) = (X r F A Ffl ~r FA.) where r is an Xl l -isomorphism and (A--~i~A.) is an F-initial A-source -- or, equivalently, l every F-sink has an (F-final A__-sink, isomorphism)-factorization. (cf. [He3]).

201

Proof: (i)

(4) ---~(i) and ~(5).

Ao-sOurce. tion

(i)

B

Numbers

'scheme of pri-

('successor')

('step function')

g~f(a,n)

= gn(f(a))),

> AxN

-:

w i t h a ter-

and a N a t u r a l

i.e. w i t h the f o l l o w i n g

('initialisation'),

A×N

A

Universes

(1)

given is

229

and w h i c h is u n i q u e in t h i s regard, then h -= g~f. As

seen

i.e.

if h: A x N - - + B

(! : A - ~ i is the u n i q u e m a p

a b o v e for the N N O - s t r u c t u r e ,

defined algebraically,

i.e.

(left projection) , A , B ~ - + r :

i n s t e a d of g~f

satisfies

(I),

into the t e r m i n a l o b j e c t i) . the other p a r t s of the s t r u c t u r e too a r e

in terms of o p e r a t i o n s

(e.g. A , B ~ - ~ A × B ,

A , B ~-+l: A x B - - ~ A

A x B - ~ B, f: C--7 A, g: C--+ A ~-~ (f,g) : C - - ~ A x B

(induced

map) ) and e q u a t i o n s and - for the N N O - s t r u c t u r e - a n i m p l i c a t i o n of equations.

2. G e n e r a l For reads:

scheme of p r i m i t i v e r e c u r s i o n

the c l a s s i c a l n a t u r a l n u m b e r s the To m a p s g: A - ~ B

(general)

('initialisation')

f: A x N - - + B such that f(a,O)

= g(a)

s c h e m e of p r i m i t i v e r e c u r s i o n

and h: A x N x B ~

and f(a,y+l)

In a P r i m i t i v e R e c u r s i v e U n i v e r s e this w o u l d read: is a s s o c i a t e d a u n i q u e f: A x N - ~ B Axs

A×N Freyd

verses

>A×N

f

> B =- A × N

shows this for topoi, ([Fr],

such that A

((l,r) ,f)

(A,O!) ; A x N h

~ (AxN) x B

'the'

like

(2) l i t t e r a l l y a s

B e c a u s e the n o t i o n of P r i m i t i v e R e c u r s i v e U n i v e r s e to g i v e a p r i m i t i v e r e c u r s i v e a l g o r i t h m w h i c h p r o d u c e s of the a b s o l u t e l y f r e e

~ t r u c t u r e E of type of P.R.

sets of sets ~nd of m a p s and w h i c h p r o d u c e s

Recursive Universe

(the L i n d e n b a u m - A l g e b r a ) ,

sets of sets and of maps.

(f: A--+ 9,g

in =

it is p o s s i b l e

at the same time a c o n g r u e n c e on 'the'

(initial)

Primitive

i.e. the f r e e one over the two e m p t y

and the m a p s of E

this s t r u c t u r e

(_E,~)

(Herbrand-model). T e r m s of the (map terms),

t h e o r e m s a r e all ('unigersal p r o -

since it is p r i m i t i v e recursive.

in [La] by w r i t i n g it up as a p r i m i t i v e

o u t - G O - T O - p r o g r a m in the sense of

(and

the sets and the m a p s

(equations) .The c o u n t i n g a l g o r i t h m

is f o r m a l i z a b l e in our system,

fact is shown in d e t a i l

(counts)

By t h e n a t u r e of f r e e construction,

(set terms)

: A-+B)

and

in the c l a s s i c a l frame.

is a l g e b r a i c ,

it into

c o n s t i t u t e s our t h e o r y as well as its c a n o n i c a l m o d e l t h e o r y a r e the sets of E

>B

U n i v e r s e over the two e m p t y

(counts)

that s t r u c t u r e w h i c h - t a k e n as e q u a l i t y - - m a k e s

This l a t t e r

r e c u r s i v e PL - w i t h -

[ B - L].

s t r u c t u r e of the N a t u r a l N u m b e r s O b j e c t N

introducing

shows f i r s t

g

B~A

his proof u s e s o n l y m e a n s of P r i m i t i v e R e c u r s i v e U n i -

Primitive Recursive Universe

After

( AxN)×B-+ B

5.22). V i e w i n g v a r i a b l e s as p r o j e c t i o n s a l l o w s us to state

3. C o n s t r u c t i o n of

4. S ~ n i r i n g

f

h:

> B.

assertions

gram')

(2)

T o any g: A--~B,

p a r t i a l l y a l s o to prove)

pairs

B is a s s o c i a t e d a u n i q u e

= h(a,y,f(a,y)) .

O+x ~ x

+: NxN --+ N and

s(x+y)

and

° : N x N --+N

_= sx+y

as u s u a l by p r i m i t i v e r e c u r s i o n one

and f r o m this that

(N,O,I,+,')

is a u n i t a r y

cc~nu ta tive semir ing.

5. L o g i c a l Freyd

s t r u c t u r e of P r i m i t i v e R e c u r s i v e U n i v e r s e s

shows in

[Fr],

5.11

that

i ~

O

s N ~--

N

c o n s t i t u t e s a sum. W e g e n e r a l i z e

230

(A,O!) this

slightly

constitutes

to the f a c t

a sum,

(flg) : A x N - - e C

i.e.

:_= (A f

which

Let us N--+N A ~+

N = A ~

f

truth values. i.e.

the

6. F u r t h e r

We d e f i n e (OIN),

showed

sum.

maps

f,g:

~ f

and

O with

h(A~s) (A,O!)

~ N

-= g

all

(O1 i) :

of t h e f o r m

predicates

junctors.

arguments

same method

:=

or

((O10!) I(O[I!)i)~

:~ N × N

logical

sg

1. M a p s

for A = I nullary

the o t h e r

of

A - ~ C. T h e m a p

signum-fu~ction

greater

and

N instead

=_ g(a) .

by truth-table

By t h e

=_ g. We u s e

The

using

boolean

uniqueness

tautologies

N,

of

on predicates

showed.

p(O)

p: N - ~ N

_= O, p(sx)

as

~ x, a n d

induced

o u t of t h e

the truncated

sum

O

1

subtraction

~ N L

s

N

by

by

p~N

N×N

> N

O'-x _= O,

:=_ N x N

> N,

i.e.

x = O ~ x, x-'sy = p(x'-y) . F o r

sx-'sy = x-'y ( c o m p e n s a t i o n ) ,

(x+y)-" (y+z) properties induction

~ x-z, of

x-'x _= O, p(x'-y)

(x-'y)-'z = x-(y+z)

equality

and

order

(association).

this

_= pxay, These

o n N to be i n t r o d u c e d

operation

we

(x+y)'-y -= x

laws

below

are

show

(absorption)

important

a n d for

for

the proof

of

pr inc ipl es.

Exponentiation

above

similarly

are

NxN

A×N

is a u n i q u e

on N

-

der.

N--+N,

there

h(A,O!)

number

:=_ (I IO!):

the predecessor

i.e.

natural

neg

the

Given

of N: T h e

predicates,

can be

Algebra

structure

A~s ~ AxN ~

(flg) (Axs)

h(a,l)

are called

junctors

sequel

and

A

g: A ~ N - - + C

and

N

(f Ig) o u t of

in t h e

_= f

=_ f(a)

every

'truth-tables', the

f: A - - > C ,

has the property

h(a,O)

identifies

Define

for

induced

p:-=

g--> C)

the 2-1ike

sg

N

by their

equations

used

A

study further

set A

2 for c a s e - d i s t i n c t i o n :


intuitively

arbitrary (fl g) (A,O!)

object

C I AxN ~

gives

for

such that

a not yet available h

that for an arbitrary

The

is d e f i n e d

equations

lemmata.

be d e f i n e d We n o w

by p r i m i t i v e

of t h e s e

Theathe

operations

notions

of

recursion, necessary

'prime

number'

likewise

in t h e and

division

sequel

are

'greatest

with

remain-

shown using

common

divisor'

the can

as usual. introduce

We will discuss

equality

on N a n d d e r i v e

an important

extension

Peano's

axioms

f~om

of P e a n o - i n d u c t i o n ,

the

primitive

recursion.

'diagonal

in-

duction' .

7. P r e d i c a t e

of

The distance

equality between

equality

by

N~N

= ~N

Equality

is r e f l e x i v e

Peano' s a x i o m s Natural

numbers

read

o n N and natural :-~ N X N

and

the proof

numbers dist>

symmetric

N and

in o u r f r a m e w o r k

are arrows

x:

i--+N.

of P e a n o ' s

is d e f i n e d neg)

N

equality

by

axioms dist(x,y)

(neg m e a n s

equality

on N implies

as follows:

:=

(x'-y)+(y±x), with

logical

O).

equivalence.

231

P1 P2

O: I--+N (x=y)

is a natural number

impl

(sx=sy) = i!

w h e r e infix n o t a t i o n is used: x = y instead of =(x,y), a impl b instead of impl(a,b), impl being the i m p l i c a t i o n junctor. P 2 m e a n s that the successor is well defined, crucial P3

(sx=sy)

i.e. that the 'map'

s has this

(internal) p r o p e r t y of a mapping. impl

(x=y) =_ l!

i.e. s is injective as a mapping. P4

neg(sx=O!)

-= I!

i.e. O is not a successor. P5

induction,

here in f i r s t order form:

If for a p r e d i c a t e (i)

X(a,O)

(ii) x(a,x)

-: i!

(this stands for 'property')

2t : A × N ~ N

and

impl ~((a,sx) -m i!

then x (a,x) =- I!

i.e.

is overall true.

Specializing to A = i g i v e s the c l a s s i c a l P5. T h e p r o o f s of P1 to P4 are straight forward by the use of tautologies,

the

lemmata a b o v e and the cited p r o p e r t i e s of equality. For proving P5, show that

) N

N

1

(X,N)

-(i,O~>

(X,N) NxN

commutes, u s i n g

_= (or (l,Xsr) , sr)

NXN

or(l,~(sr) = and(or (l,~sr),impl()f,)/s)) . T h e n c o n c l u d e by

u n i q u e n e s s of the iteration m a p above. Sometimes we u s e a sharper induction scheme, the / ~ - i n d u c t i o n : If

X(a,O)

Herein

_= i! and ( i=O ~(a,i)) impl )~(a,sx) ~ I! then x i/__kO~(a,i) is g i v e n by iteration of and: NxN--~N.

)£(a,x) -= l!

T h i s induction p r i n c i p l e is proved by a p p l i c a t i o n of the foregoing principle.

8. D i a g o n a l induction O n l y in special cases the induction p r i n c i p l e s so far d i s c u s s e d are a p p r o p r i a t e for showing g e n e r a l t r u t h of n - a r y p r e d i c a t e s on N. In particular t h e y d o not suffice for E l ~ n e n t a r y Arithmetics.

We need another

induction principle:

'diagonal induction' ,

232

i.e.

induction

along

If a p r e d i c a t e

the direction

of t h e d i a g o n a l

of N x N w h i c h

of t w o N N O - v a r i a b l e s

76: A x ( N x N ) --~N

~ (a, (x,O!))

-= I!

i.e.

X is t r u e on t h e x - a x i s ,

(ii)

~i(a, (O!,y))

=

i.e.

X

(iii)

~i (a, (x,y))

is t r u e o n t h e y - a x i s ,

]!(a, (sx,sy))

impl

-= i!

as f o l l o w s :

satisfies

(i)

i!

reads

i.e.

truth

is o v e r a l l

true.

and

is i n h e r i t e d

in p a r a l l e l

to

the diagonal, then

X ( a , (x,y))

Geometrically, that NxN

i.e.

this principle

an

to p r o v e

isomorphis~n

the rationals

it.

~

is o b v i o u s .

is w h a t w e u n d e r s t a n d

far developped define

-= I!

B u t s i n c e we c a n n o t

geometrically

In o r d e r

count:

to r e d u c e

N--*NxN

be

sure at the moment

by it, we h a v e to u s e the two dimensions

in a n a l o g y

our A r i t h n e t i c s to one,

so

we f i r s t

to t h e u s u a l m e a n d e r - c o u n t i n g

of

by

s

N

~

o/ ~

, I

=-

1 Ic ° u n t

=

N

NxN

I

NxN

(N×N, nc~ l)

;~

Icount I

(N×N)x N

(pxsl ( s r , 0 ! ) l)

-~ NxN

I (N×N) x s (N~N)x N

i.e. b y count(O)

-= (0,0) if 1 c o u n t (n) _= 0

(s r c o u n t ( n ) ,0) c o u n t (sn) =_ (i~ 1 c o u n t ( n ) , s

and

t r y to

show that count count

where

h a l f (0)

First

we

For

:= O,

-i

h a s as a n i n v e r s e (x,y)

h a l f (sx)

show that count

this commutativitiy

-i

of

r count(n))

count

-i

: N×N--+N

d ef ined b y

:m h a l f ((x+y) • (x+sy)) +y

:- h a l f (x) + o d d (x) a n d Odd (0)

.

is a r e t r a c t i o n (~)

else

in

for c o u n t .

:- O, o d d ( s ~ )

:_= neg odd(x)

233 N

)

~count

~

0

Icount

.-=

> N×N ~ e o u n ~ N-1 ~ N , n e g i)

(N×N) xN ((p×s l(sr,O!)l)

~

N

> NxN count -i

(~)

N

~

N

is sufficient by u n i q u e n e s s of the iterated map. B y pulling b a c k the sum the upper edge of of

(s,O!)

and

(~)

(NxN,O!)

N~N

-> (NxN)xN
N~N ~

sxN

NxN

((O],s) Isxs), the latter being an induced m a p out of N x N d e c o m p o s e d

the same way. N e x t it is shown that the upper part of of

half((sx). (ssx))

and h a l f ( ( s x + s y ) . s ( s x + s y ) ) + s y . half ( (x. sx) +sx) Corresponding

(~e)

and the induced out of the

is the induced out of the sum a b o v e (same)

sum of

half((sx)- (ssx))+sx

S i m i l a r l y one shows that the lower p a t h is induced by

and the induced of

half ( (sx. ssx) +sx)

and

half ((ssx+y) • (ssx+sy)) +sy.

inducing c o m p o n e n t s of upper and lower p a t h a r e equal: this is immediate

for c o m p o n e n t s two and t h r e e and v e r i f i e d by d e f i n i t i o n of half and d i s t r i b u t i v i t y of '. ' over '+' for the c o m p o n e n t one. It would be sufficient to show that count

-I

is a section by giving a n a r b i t r a r y

r e t r a c t i o n of it. A proof using count seems not to w o r k v e r y well. We f o l l o w instead [Pe], [H-B] and

IDa] by u s i n g the integer

ted by root(x). A b b r e v i a t e

q(z)

(part of) r o o t of a natural number, d e s i g n a -

:= half (root (8z+l ) +l) "-I

q' (z) := 2z'- (q(z)) 2 Then, as in the l i t e r a t u r e cited count(z)

:= (q(z) '-half(q' (z)'-q(z)) ,half(q' (z)-'q(z)))

is the c a n d i t a t e for the wanted r e t r a c t i o n of c o u n t proof is the e q u a l i t y

q(count-1(x,y))

-i

. The c r u c i a l point in Davis'

= half (root(2x+2y+l) 2+8y) +1) "-I = x+y.

If we can show this e q u a l i t y in P r i m i t i v e R e c u r s i v e U n i v e r s e s we are done,

because

from this the r e t r a c t i o n p r o p e r t y of count is straight forward. The wanted e q u a l i t y is further r e d u c i b l e to

root( (2x+2y+1)

2+8y)

=

I 2x+2y+1

if

[ 2x +2y+2

otherwise

2y ~- 2x+l

T h i s is c l a s s i c a l l y ~evident' ,and we have a c l a s s i c a l proof for it using nested induction.

234

Diagram

f o r m of the last statement:

((x,y) , sg (2y'- (2x+l))

N×N

(2x+2y+l] (2x+2y+2)l)

(NxN)x N

> N

/

\ r o o t ((2x+2y+1) 2+8y)

of this the

We will call fulfillment

'root property'

for a P r i m i t i v e R e c u r s i v e

Universe. For the d e f i n i t i o n [H-B]: The

of r o o t there a r e at least two p o s s i b i l i t i e s .

root(O)

- O,

second one - used

which

is c o m b i n e d

root(sx)

in our proof

- O,

rrem(sx)

-~

(categorical

translation

So, f r o m t h e

'root property'

By the

induction

T h e proof of

taking

statement

the integer

- is the f o l l o w i n g

p a r t of the root:

~ O

s root(x)

if

root(x)

otherwise

rrem(x)

O

if

s rrem(x)

otherwise

by using

The first f o l l o w s

root(x))2"-sx)

rrem(x)

= 2.root(x)

= 2.root(x)

sums)

follows

N ~= N×N. F r o m

this we p r o v e the p r i n c i p l e

of

as follows:

/k-induction X-= I[

by

rrem(O)

r o o t (sx) =

yields

of the c l a s s i c a l

w i t h the r e m a i n d e r

root(O)

diagonal

=_ r o o t ( x ) + n e g ( ( s

principle

we s h o w

)~(Axcount)

_= l! r count being a n i s o m o r p h i s m

w h a t we h a v e to show.

)6(A×count)

_= I!

turns out to be quite a d i f f i c u l t

one.

It u s e s the

u n i q u e n e s s of the induced m a p o u t of the sum -i count (O! ,N) count -I (SxN) N -> N T 1 q A

(A,Pi:A--+ A i)

k:TiB ÷ TIqA

T l q P i ~ T 1 q A i = Tlh i

s.t.

in

Az ,

240

we have to look for a unique qpioh

= h.~

~:~q ÷ 1

for each

i.

h:B + q A

Consider

with

the following

is given by the condition

a Tl-final

T1h = k

£q < 1

sink given by codenseness

of

and

diagram where

and

(mj,B)j

is

E1

T2P i = Tlqp i T2A

I

~

> T2A i

k

T2E2C j

1 = T2EAi

> TIB

~ T2A i

T2£m j = Tim.J Here

r.]

(for each

T2-initiality provides

T2Zh i = Tlh i

of

j 6 J)

is the lift of

(A,Pi).

a unique

h:B ÷

Finality qA

with

k~Tlm j

of the sink Tlh = k

and

due to

(mj,B) hom.

now

= qr..

J

To get

qPi°h

= hi

By the following Tlq(Pi°rj)

it is sufficient

proposition

= Tlq(~ij)

Pi°r'J = m..ij, hence

To prove

uniqueness

we have

Tlh = k

we have

to prove h.om. 1

j

By faithfulness qpiohom.j

of

h,

and

= qPi°qrj

assume

qpi°h

= h.1

will have the same final lift by

of

J

qPioh~ m. = h.om..

= q(~..)

and h~nce

1j

T2

we get

= h'°m''l ]

that for some for each (m.,B)

as

[:B +

qA

i.

Then

Tlh

k,

by a similar

J

argument Theorem T2°~

as above,

hence

[8] we conclude

= T I.

h = h. that

q

Now apply proposition

Again by Tholen's

taut

has a left adjoint

~

1.

construction

We go on establishing hypothesis

mentioned

this proposition

covers

in the introduction.

some conditions

in the preceding

with

[]

We will point out later to what extent Dubuc's

lift

equivalent

proposition

that

to the crucial

£E 1 = E 2 .

241

Proposition

3 :

corresponding lowing

(i

q

coarsest

conditions

~

(ii

Let

are

T2h (iv)

diagonal

of d i a g r a m for e a c h

(i)

faithful. the

h:EIC--+qB

C c

and

(iii)

underlying

(i) =

(iv)

:

(~B)

from

the

Hence

we

ob C

and

(ii) and

are

(iv)

functors

Given

(iv) ~ phisms ce

get to

(i) :

the

the

fol-

h:E2C

some

÷B

with

which

1

are

equivalent

are e q u i v a l e n t

Ti

Zq
0 Corollar~.

for any vector

X >i0 , X # 0 .

If S is an excellent Gabriel ~-species and S' is a subordinate of

S, then S' is an excellent Gabriel F-species. For

exanple, the Brenner quadratic form of the species 0 5 :

i~4~5 (where ====~ denotes an arrow with capacities 2,1 and both capacities i) is the following

:

~

denotes an arrow with

275

2

2

2x21 + 2 x 2 + 2x~ + x 4 + x 5 - 2xlx 2

--

2x2x3

2x3x5- 2xlx 4 - x 4 x 5 + 2xlx5 .

- -

Writing this quadratic form as a s~n of squares :

¼(2X1

2X2 + X5)2 + I(2XI

2X 4 + X5)2 + ~(X 2

2X3 + X5)2 + ix2

, it is obvi-

ous that 0 5 is excellent. 2. Representations of F-species Let F be a division ring and let S be a Cxabriel F-species. A representation

(V) of the species consists of :

a) a vector space Vx, such that

dimF

V x = dx < ~

, for every xeG ;

X

b) an Fa (y)-linear map

%

: Vs(y ) ~ M ~ F s (y)Y

Va(y ) , for every arrow yeF .

A morphism (a) from the representation (V) to the representation (W) is simply a family [ ~ x~G ' ~x being an Fx-linear mad V x ycF,

~ a ( y ) % = Wy(~s(y)~l) Denote

~(S)

~ Wx, such that for every

"

the abelian category of all representations of the Gabriel

F-species S . Remember that, given a species S, for every d-cyclic subgraDh of the form (I) there exists an isomorphism

i : M y ~ . . . ~ M y . ~~ ~

My0.,,~...~MVm "

Definition 4. A representation (V) is called commutative if for every d-cyclic subgraph of the type (I) we have : Vyn,O ( % n _ l ~ l ) ~

... o (v~vu' ~ l ~ ' ' ' ~ l )

= W..O(VmYm-l"~ l ) a

.... (%~i~...~I)6

o(l ®i) , where

i

is the iscrnorphism for this subqraph, chosen in the definition of S .

Denote by

~

(S)

the full subcategory of

~

(S)

whose objects are the

commutative representations. Then, obviously, R_~ (S) is an abelian cateqory and every object (having a dimension) d e ~ s e s

as a finite direct sum of indecc~po-

sable objects. This decomposition is unique. 3. Commutative representations of the species 0 5 The F-species 0 5 is based on the following valued graph (the heights are indicated above each vertex) : If F is contained in another division ring G, such that

dim F G = dim C~F = 2

2

2/2 1

2

----~3~.. 1 ~ 5

and G is a special

G-F-bimodule, then the admissible modulation of the species is the following : A cc~mutativ~ representation (V) of this species is ognpletely described by the following ccmanutative diagram of F-vector spaces :

G

G ,G

276

D

~here ~

is the F - v e c t o r

space structure o b t a i n e d

ture o f X b y restriction of scalars; If E = 0, then thermore,

from the G - v e c t o r

the maps a F = a and b F = b are C~linear.

(V) is in fact a representation

if A = 0, then

(V) is a representation

of the valued graph C 4 ; fur-

of the v a l u e d qraph F 4 . These

trivial cases are e l u c i d a t e d by Dlab and Ringel in [4]: there exists indeconposable

co~utative

Proposition

representations

deccmposable

commutative

Then the F-species

representation

F

F2

G

w h e r e i and r are the canonical maps

are, r e s p e c t i v e l y , (i,i,i,2,1)

(from section j (i) =

rV

i), ~ is the p r o j e c t i o n on

(l,y). The dimensions o f these re-

:

, (1,1,1,2,2)

, (1,1,2,2,2)

and

W e sketch the proof, w h i c h contains several stages; deccr~oose

(V) as a direct stun (V')(~(V")

Ix)sable and c o ~ u t a t i v e . -

-

-

-

Ker (d) (] Ker (ba) : 0 ; = 0 ;

Ker (d) (] Ker (a) = 0 ;

- Ker(a) ~ -

-

-

-

Ker(d)

Ker(ba) C

Ker(d)

a(Ker(d))

= 0 ;

Ker(ba)

;

= 0 ; Ira(b) ;

- Ker(c) C

I_m(ba)

-

;

Ker (ba) r~ Ker (d) = 0 ;

- Ker(c) C

Ker(c)

= 0 ;

(1,2,2,2,2)

.

at each sta~e w e try to

, (V) b e i n g supposed nontrivial,

indeccm-

In this m a m m e r we obtain the followin 9 conclusions

Ker (d) ~3 Ker (a) = 0 ;

Ker(d)

F2

r~

the second co _r~ponent and j is given by

(i,i,i,i,i)

(that is E # 0 and A # 0) i n d e x -

F2

G

presentations

= 2

is of one of the followin~ five types:

F

G

dim F G = dim ~

05 has exactly 41 types of in-

representations.

Proof. W e e s t a b l i s h that any nontrivial sable c ~ t a t i v e

36 types of

w h i c h are trivial.

2. Let F c G be division rinqs such that

and G is a special G-F-bimodule.

space struc-

;

:

277

- Ker(b)

= 0 ;

-

Ker(e) ~

-

E

=

Im(d)

~(e)

;

.

A t this stage, denote b y K the intersection i) First, Denote

X = ba(Ker(d));

in Ker(c).

Then

then, obviously,

Ker(c)

= X•Y

and

XC

= a ( ~ )

, S" = bl(c"),

qhen

= E'

, e(D")

where A' = Ker(d')

Supposing that i-i)

(V) =

A' = K e r ( - ~

= E"

and Ker(d")

Ker(c).

andwe

.

Im(ba)

and D = Ira(d).

Cheese a cc~plement Y of X Denote C' = X and let C"

. Denote E' = c(C'), E" = c(C"), B' =

, A" = al(B"), decompose

D' = d(A'),

D" = d(A")

(V) as a direct s u m

(V')~(V"),

= 0 .

(V) is indecomposable,

(V')

Ker(c) ~

X = ba(~----~).

he a complement o f C' in C, c o n t a i n i n g Y

e(D')

Ker(c)~Im(ba)

suppose K # 0 ; then it follows

, that is, ~

that, in this case, the r e p r e s e n t a t i o n G

I /

w e n e e d to consider two possibilities.

= A and ~

= C . It follows e a s i l y

(V) has the following tvoe

:

1 ---------- G

"~'~

G /

F

-'>-r --.. F 1-2)

(V) =

(V") , that is Ker(d)

let C" be a c ~ l e m e n t

= 0. In this case, denote C' = ~

of C' in C. Denote A' =

(ba)-l(c ') , B' = a(A'), D' = d(A')

E' = c(C') a n d a n a l o g o u s l y A", . . . .

It follows that

s~n

and

(VI)~(VI) f

"

,

with

C'

=

~

Ker(c")

A g a i n w e n e e d to consider two possibilities 1-2-1) r~nsion

(V) =

(V{)

, that is

Ker(c)

and

=

(V) is d e , r e p o s e d as a direct 0 .

:

= C. In this case

(V) has the type o f di-

(1,1,1,2,1).

1-2-2)

(V) =

isc~orphisms.

(V~') , that is Ker(c)

The representation

2) Suppose n o w

= 0. In this case a,b,c,d and e are all

is of the type o f dimension

K = 0 ; then, obviously,

Ker(d)

= 0 and, because Ker(e)

contained in Ira(d), it follows inmediately that also Ker(e) In the next stages w e obtain -

Im(ba)

-

C

=

~

Ker--6~7~7

- dR

(1,1,1,2,2). is

= 0 and D = Im(d).

:

; ;

C = 2dim G Im(ba)

;

- dimG A = 1 . There are two possibilities, Im(ba).

debendind on the coincidence

It follows n o w easily that these two p o s s i b i l i t i e s

of dimensions

(1,1,2,2,2)

4. Extensions

and

(1,2,2,2,2).

The p r o p o s i t i o n

o f ImCo) w i t h

c o r r e s p o n d to the types is proved.

of a t h e o r e m o f Gabriel

The result of Gabriel about the representation

type o f a graph is well-known:

278

Theorem 1 - .positive form (Gabriel,[6]). Let (G,F) be a graph and F a division ring. The category

Re_F(G,F)

is of finite representation type (that is, it

has only a finite number of types of i n d e ~ s a b l e if and only if

i) A n (n >i i) List 1

objects), independently on F,

every connected subgraph of (G,F) is contained in the list i : 1--2--...--n 21 ~ 3 - - . . . - - n

2) D n (n >i 4)

; ;

3) E n (6~n, , by [14~. (1) ~ (2) i s easy. (2) ~ ( 3 ) : a) F i r s t , l e t us suppose t h a t k i s a f i n i t e tarset B ~c~, denote by mb:b ---> c the morphism of k, b ~ B . We show that Set k fulfils the Tarskl cube property. Let F:k~-->Set be an object, let ~'~:F--~ be a natural equivalence. If x~_F(c) (or x ~ F 2 ( c ) ) denote by ~.(b,x) (or ~(b,x)) the cardinality of the preimage of x in F(m b) (or F2(mb) , respectively). Since F~---~ 3, one can construct, for every element x ~ F ( x ) , an element ~ F ( o ) such that 0 Set t h e £ u n c t o r such t h a t Xn(n) = ~ 0 , i ~ , Xn(C) = ~0~ = Xn(b) for all b e co\~'n~. Xn(m b) is a constant map for every b e co obviously. We show that ~ = = %X !n e co } is c~-oroductively independent. If f: ~ - - - > Set the functor such that Xn(a) = ~O,l,...,pn} , Xn(b) = {0,I~, Xn(C) = {0~, Xn(m b) sends 0 to 0 and every point of ~l,...,pn ~ to i, Xn(m c) (and Xn(~) if b < c ) is constant. We show that ~ = ~Xn~ n ~ c ~ is cD-~roductively independent. If f: ~ > co is a map and Af = ~ c a ~ n , then f can be recognized from Xf because, for every n ~ co , f(n) is the number of ell x such that its preimage in Xf(m b) consists of Pn points. (Indeed, any such point x has necessarily all the coordinates equal to 0 except one which is equal to i; and this coordinate corresponds to a copy of X n in the product Xf.) If A c ~ end X A is as in 11.2, we decompose the functo~ X A on subfunctors corresponding to the points Y e X A ( C ) and recognize every l e A from these

eXf(b)

subfunctors.

307 c) Now, we f i n i s h t h e p r o o f o f t h e t h e o r e m . I f a p o s e t k c o n t a i n s k 1 o r k 2 , t h e n e v e r y f u n c t o r Xn, d e s c r i b e d i n e) o r b) c a n be e x t e n d e d t o k s u c h t h a t we o b t a i n a n ~ u - p r o d u c t i v e l y independent collection of o b j e c t s i n S e t k . I f k c o n t a i n s n e i t h e r k 1 n o r k 2 ( w i t h b < c o r b and c b e i n ~ i n c o m p a r a b l e ) , t h e n e v e r y i t s component i s n e c e s s a r i l y s finite target. 7. Theorem. Let k be e poset such that there exists an object in it, in which three distinct arrows iniciate. Then every commutative semi~roup has 8 productive representation in Set k. Proof. Since Set k is cO-distributive, it is sufficient to construct, for every infinite c a r d i n a l o~ , s n c ~ - p r o d u c t i v e l y independent coll e c t i o n o f o b j e c t s o f S e t k. L e t t h e t h r e e d i s t i n c t a r r o w s o f k be rob:S---> b, m c : a > c , m d : a - - - > d . Then t h e f u l l s u b c a t e g o r y g e n e r a t e d b y ~ a , b , c , d } h a s one o f t h e f o l l o w i n g f o r m s .

h of k

Hence we may suppose that there is no arrow from b to c and from b to d and there is no arrow from c to d. Let ~ { 3 I ~ e c6 ~ be s collection o f d i s t i n c t cardinsls. We d e f i n e a f u n c t o r Xt3 : h > Set by

X~ (a) =~O,l]u('r~×~}),

X~(,b) =~0}, X~(c) = ~O,l}, X~(d)

= ~0,I,2~,

X/~ (m O) sends O to 0 and all the other points to I, X ~ ( m d) sends 0 to O, 1 to 1 and (x,2) to 2 for every x e ~ a n d , i f t h e r e i s an a r r o w d > c i n h , i t s X ~ - i m a g e s e n d s 0 t o 0 and both I and 2 to i (for the arrows ending in b, their X ~ - i m a g e is evident). We prove that ~ X ~ I ~ eoc~ is sn oc-productively independent collection of objects of Set h . First, we show that every f e co~

can

be recognized from Xf = /~TTocxf(~). For every z ~ X f ( c ) denote M z =

=

(Xf(ma))'l(z) and put L = ~ z e X f ( c ) I card (Xf(ms))(M.) = 2~. One can verify that, for every z eXf(¢) = ~ o c ( ~ O , l } f ( ~ ) ~ , z~L

iff all its coordinates except one are equal to O.

Hence f ( ~ ) is the number of all z e L with card M z = P//3 ÷ I. Now, let A c co ~ , card A = c¢ , X A be as in 11.2. We show that A can be recognized from X A. Indeed, consider all subfunctors of X A determined by the points of XA(b). Each of them is of the form Xf end £ can be

308

recognized from it, hence A can be recognized from X A. Finally, e v e r y f u n c t o r X~ :h ~ S e t can be e x t e n d e d t o a f u n c t o r k ~Set such t h a t t h e c o l l e c t i o n o f t h e e x t e n d e d f u n c t o r s i s an o ~ - p r ~ u c t i v e l y independent

collection

of objects

i n Set k.

III. Pr_oductive representation with a ~iven subob~ect I. Let o¢ be an infinite cardinal, let ~C be an ct_-distributive category. Let ~ = ~ X / ~ i ~ e ~ b e a collection of its objects, let Xf and X A be as in II.2. We say that an object Y of ~C is ~ - s o f t if, for every A,B c co ~c with card A = ~ = card B, x s iff

x,

x B.

2. Theorem. Let oc be an infinite cardinal, ~C be an c~-distributive category, let ce = { X ~ i ~ e oc~ be an ~.-productively independent collection of its objects such that S ~ ( T , X ~ ) ~ ~ for every ~ ~ oc ,where T is a terminal object of JC . Let ~L be 8 class of monomorphism~ closed with respect to the composition and the forming products and containing all coproduct-inJections and all morphisms inioisting in the terminal object T. Let ~cc be as in 11.2. Then, for every ~ soft object Y of ~C , there exists a productive representation ~Ys ~s ~ ~oc ~ of ~ such that Y is an ~t-subobJect of every Ys" Proof. For every A ~ So~ put rA = A ~

Y k × XA"

Then Y is an ~t-subobJect of every YA a~d ~YA IA ~ ~oc ~ is a productive representation of ~ c • Indeed, YA K ¥B ~ YA+B (the reasoning is similar to 11.3); since Y is c~-soft, X A is isomorphic to X B whenever YA'~--YB; but then A = B because c6 is c~-productively independent. 3. In a lot of concrete cases mentioned in 11.4, the co-productively independent collection ~ is constructed such that every object of the category in question can be embedded in a q - s o f t object. This is fulfilled also for the co-independent collections of objects in Set kl , setk2, constructed in the proof of Theorem 11.6. For example, if T :k1 ~ Set is given, we embed it in a functor Y:kl---> Set such that the preimage in Y(m b) of any x e Y(c) is infinite (for every b eB). One can verify easily that X A can be recognized from ~ = = A ~ c ~ y k ~ x A if we consider only subfunctors of ~ , determined by the points x 6 ~ (c) which have finite preimages in the ~ (m b)'sAn analogous reasoning works for k 2 as well. Hence we can enrich

309

Theorem I I . 6 by t h e f o l l o w i n g a s s e r t i o n s , e q u i v a l e n t t o t h e o t h e r s . (4) E v e r y f u n c t o r ~ : k - - , S e t can be embedded i n t o s f u n c t o r i s o m o r p h i c t o i t s cube but not t o i t s s q u a r e . (5) E v e r y c o u n t a b l e commutative s e m i group has a p r o d u c t i v e r e p r e s e n t a t i o n i n S e t k by f u n c t o r s , c o n t a i n i n g a g i v e n f u n c t o r ~ :k---> -- - ~ S e t . A l s o , i f k i s a p o a e t w i t h an o b j e c t a i n which t h r e e d i s t i n c t arrows i n i o l a t e and eeoc= ~X~ I ~ r=~_~ i s t h e ~ - p r o d u c t i v e l y i n d e p e n d e n t c o l l e c t i o n o~ o b j e c t s o f Se t k, c o n s t r u c t e d i n I I . 7 , t h e n e v e r y o b j e c t o f S e t k can be embedded i n a c e ~ - s o f t o b j e c t , hence e v e r y commutative s e m i g r o u p has a p r o d u c t i v e r e p r e s e n t a t i o n i n S et k by f u n c t o r s c o n t a i n i n g 8 g i v e n f u n c t o r ~" :k 4 S e t . IV. How l a r g e a r e t h e r e p r e s e n t i n g

objects?

1. I f we i n v e s t i g a t e p r o d u c t i v e r e p r e s e n t a t i o n s i n a c o n c r e t e c a t e g o r y , t h e r e i s a n a t u r a l q u e s t i o n : how l a r g e a r e t h e u n d e r l y i n g s e t s o f t h e r e p r e s e n t i n g o b j e c t s . I n many c o n c r e t e c a t e g o r i e s , t h e f o l l o w i n g e a s y m o d i f i c a t i o n o f t h e b a s i c method p e r m i t s t o d i m i n i s h t h e c e r d i n a l i t y o f t h e u n d e r l y i n g s e t s : one c o n s t r u c t s t h e c o l l e c t i o n c~ = ~ X ~ ~ ~ e o c ~ such t h a t any X~ c o n t a i n s e d i s t i n g u i s h e d p o i n t , say ~ . , . a n d , f o r any f ~ ~ , Xf i s not t h e whole p r o d u c t ~I'~e~cx f ~ J as in II.2, but only its subobJect consisting of ell those points, which differs from

~f = B e~Yoc ~f(~) in at most finitely

many coordinates. The collection ~ and the distinguished points ~/~ , ~ oc , have to be constructed such that any set A c c~°c with card A --~ oc can be recognized from the object X A being a coproduct o f t h e s e new X f ' s . 2. If this modification is combined with the application of Theorem Xll.2, one can obtain, for example, the following assertions: every countable commutative semigroup has s representation by products of a) b) c)

countable topological spaces, containing a ~iven countable space (see[20]), countable posers, ~raphs, tolerance spaces, containing a given countable poset, graph, tolerance space (see [17~), countable unary algebras, containing a given countable unary al~ebra.

S. Let us show an application of this idea on the category (Set~) k, where Setco denotes the category of all countable sets (and k is a poser)° We prove the following

310

_Proposition. The assertion (6) below is equivalent to (1)...(5) in 11.6 and 111.3. (6) Every countable commutative semi~Toup has a productive rePresentation in (Set~) k by functors, containir~ a given functor ~J{: :k ~ S e t ~ . Proof. If • poser k contains k I (or k 2), define X n as in I1.6. Let c~n be its suhfunctor, sending any object of k I (or k 2) to a onepoint set, namely O-n(p) = tO} for every object p of k I (or k 2) and define Xf to be the subfunctor of ~ = ~n n such that, for every object p of k I (or k2, respectively), X~(p) consists of those x e f(n)( p) "at most in finitely coordie ~ ( p ) , which differ fr om ~ O ~ n nates. Then X f x X is still isomorphic to Xf.g end f still can be recognized from Xf ~by the same reasoning as in II.G°). The rest of the Proof is the same ee in II.~. end III.3.

4. Remark. L e t us m e nt i on one t r i c k more, which p e r m i t s t o o b t a i n the following assertions: every countable graph (poset, tolerance space, unary algebra, topological space) can be embedded into a countable graph (poser, tolerance space, unary algebra, topological space) which has 2 o nonisomorphic square roots [17], [18]) and also the assertion if a poser k contains k I or k 2 from 11.6, then every fUnctor ~" :k--,Seto~ is a subfunctor o£ some X:k---> Set~o , which has 2 xo non equivalent square roots. This follows immediately from the above results and the Proposition. The semigroup 5~o~ of all countable subsets of co ~ contains a subset T such that card T = 2 ~° and s + s = s" + e" for every s,s'e T° Proof. Let S be a semlgroup with a countable set of generators, say {shin 6 60 ~ and defining equations s n + s n = s n + an. for ell n,n'~ c~ • By[14~, there exists e disjoint homomorphism h:~ ...... ~ 5 ~ i.e. h(s)~h(s') -- ~ whenever s=@s ". Pu~ T : {~_~eAh(Sn)IA ~ , A4=~,

t h e n T has t h e r e q u i r e d p r o p e r t i e s . References

1. 2°

J . Ad~mek, V. Koubek. On • r e p r e s e n t a t i o n o f eemi~roupa by p r o d u e t s o f a l g e b r a s and r e l a t i o n s , C o l l . Math. 3 8 ( 1 9 7 7 ) , 7 - 2 5 . J° A d ~ e k , V. Koubek, R e p r e s e n t a t i o n o f o r d e r e d commutative s e m i g r o u p s , C o l l . Math. Soc. J e n o s B o l y s i 20, A l g e b r a i c t h e o r y o f s e m i ~ r o u p e , Szeged 1976, 15- 31.

311

3. 4.

5. 6. 7. 8o 9o i0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

J. Ad~mek, V. Koubek, V. Trnkov~, Su.~ of Boolean spaces represent every group, Pacific J. Math., 61(1975), i-7. A.L. Corner, On a conjecture of Pierce concerning direct decomposition of Abelian groups, Proc. of Coll. on Abelisn groups, Tihany, 1963, 43-48. Sur l'~quation n = pour types d'ordA.£. C.R. DavisAcsd. (Morel),, des re, Sci. Paris 235(19521 ~, 924-~926. R.H. Fox, On a problem of S. Ulam concerning Cartesian products, Fund. Math., 34(1947), 278-287. W. Hanf, On some fundamental problems conceding isomorphisms of Boolean algebras, Math. Scand. 5~1957), 205-217. J. Ketonen. The structure of countable Boolean algebras, Annsls of Math., I08(1978), 41-89. L. Lov~sz. Direct product in locally finite categories, Acts Sci. Math., 3311972), 319-322. A. Pultr, Isomorphism types of objects in categories determined by numbers of morphieme, Acts Sci. Math., 35(1973), 155-160. A. Tsrski, Cardinal algebras; with an appendix by B. Jonsson and A. Tsrski, Cardinal products of isomorphism types, New York, 194~ V. Trnkov~, X n is homeomorphic to X m iff n..4n, where ~ is a congruence on natural numbers, Fund. Math. 80(19~3), 51-56. V. Trnkov~, Representation of semigroups by products in s category, J. Algebra, 34(1975), 191-204. V. Trnkov~, Isomorphism- of products and representation of commutstive semigroup, Coil. Math. Soc. Janos Bolysi 20, Al~ehraic theory of semigroups, Szeged 1976, 657-683. V. TrnkovA, Productive representations oi" semi~Toups by pairs of structures, Comment. Math. Univ. Carolinae 18(1977), 383-391. V. Trnkov~, Cstegorial sspects ere useful for topology, Lecture N. in Math. 609, Springer-Verlag 1977, 211-225. V. Trnkov~, Cardinal multipllcstion of relational structures, Coll. Math. Soc. Jsnos Bolyai 25, Algebraic methods in Graph theory, Szeged 1978, 763-792. V. Trnkov~, Homeomorphisms of products of spaces (in Russian), Uspechi Math. Nauk 34(1979;, vyp. 6(210), 124-138. V. Trnkov~, Homeomorphisms of powers of metric spaces, Comment. Math. Univ. Carolinse 21(1980), 41-53. V. TrnkovA, Homeomorphisms of products of countable topological spaces, to appear. S. Ulam, Problem, Fund. Math. 20(1933), 285. J. Vin~rek, Representation of countsble commutstive semigroups by products of weakly homogeneous spaces, Comment. Math. Univ. Carolinae 21(1980), 219-229.

CATEGORIES

O F KITS,

COLOURED

GRAPHS,

AND GAS~S

by Antoni

O.

Introduction The main

category

games

between

games,

there them.

an a b s t r a c t

game which

combination

of t w o

automaton Section

and

different one may these

shall

consider

deal here with

between

prove

by Section

graph.

e.g.,

confine ways

categories

ourselves

to

of d e f i n i n g

be way

games,

notion

of

case of the

shown

in

at l e a s t

16

and consequently

of abstract in g a m e

some g e n e r a l

games,

of an output-state It w i l l

abstract

and

a category

the g e n e r a l

in a n a t u r a l

to b e u s e f u l

4 where

of g a m e s ,

as a p a r t i c u l a r

the n o t i o n

to d e f i n e

16 d i f f e r e n t may

natural

of a coloured

of morphisms

categories

if w e

various

notions:

of applying

noncooperative

However,

may be considered

simpler

it is p o s s i b l e

types

illustrated

We

sorts

of g a m e s ,

of c l a s s i c a l

etc.

a method

are many

categories

are a l s o

the n o t i o n

3 that

There

many

a category

of dynamic type,

is to o u t l i n e

theory.

define

games,

of one

morphisms

this p a p e r

in g a m e

one may

of two-person a category

and preliminaries

aim of

theory

accordingly

Wiweger

games.

theory

as

constructions

Some of

it is of p r o d u c t s

and

coproducts of abstract g a n ~ s a r e descri0ed, and b y S e c t i o n 5 w h e r e t h e i n t e r p r e t a t i o n of these constructions i n t h e p a r t i c u l a r c a s e of t h e t w o w e l l - k n o w n t w o - p e r s o n games isgiven.

if

We use

the t e r m i n o l o g y

(At) tC T

is an i n d e x e d

the d i s j o i n t also write identify same

s u m o f the

If

A I + A 2.

TO a v o i d sets w i t h

shall

f :A

, B

A x B

. B denotes

P o w (A) Pow+ set

tacitly

of

o f sets,

At .

Instead

cumbersome their assume

[I]

and

then of

In p a r t i c u l a r , will

StE{I,2}A t

notation

images

[5].

StcTA t

we

in d i s j o i n t

if n e c e s s a r y

that

denote

we

shall

shall

sometimes

sums;

for the

the s e t s

in

are disjoint.

the r e s t r i c t i o n and

sets

considered

reason we

question

and notation family

functor

of

f

will the

denotes and

is a f u n c t i o n

Pow

to

C.

and

C c A,

The canonical

be denoted

by

pr I

then

fI C

projections

and

pr 2

will

denote

A x B

, A

respectively.

set o f all r e a l n u m b e r s . the p o w e r denote

respectively.

set of

A.

the c o v a r i a n t

and

the c o n t r a v a r i a n t

power

313

I.

Output-state

automata,

An output-state

(I)

K =

where

A,X,Y

Y.

are

Every

next-output

A kit

(of s t a t e s , Y

to

output-state 1

A,

inputs, and

automaton

and the

1

and outputs,

respectively),


(I)

is a M e a l y

next-state

[4])

function

is a n o u t p u t - s t a t e is the c a n o n i c a l

from

A x X

automaton

function

automaton

projection

any output-state

automaton

(I) w e

= { (x,x') E X x X I V a E A k ( a , x )

It is o b v i o u s

that

A monokit relation

on

~

with

defined

the

as

(I)

onto

such

the

that

first

the

axis,

(2)

M =

where

X

(cf.

graph


and

from

the d

from

G =

that

graph.

relation

such

that

on

X. is the i d e n t i t y

is a t r i p l e

D

(of v e r t i c e s into

the

set

(x,x') C p(d)

x

to

and colours

respectively),

P o w ( X × X)

means

that

of all

there

subsets

is an a r r o w

and of

of

x'.

is a 7 - t u p l e

(A,X,Y,~,I,D,p)

UIG =

(A,X,Y,~,I)

The pregame

condition

sets

condition

A pregame

(3)

is a k i t K

}.

(X,D,p),

are

colour

= k(a,x'

is a n e q u i v a l e n c e

[4])

D

X x X;

define

X.

A coloured

such

and games

z o I = p r I. For

p

graphs,

~ = ~ 0 I.

(cf.

next-state

sets from

function

the c o m p o s i t e

i.e.

coloured

is a q u i n t u p l e

(A,X,Y,~,I) ,

is a f u n c t i o n to

kits,

automaton

(3)

is a k i t a n d

is r_egular if

U2G =

for all

(X,D,p)

is a c o l o u r e d

Xl,X~,X2,X ~

(Xl,X ~) C < U I G > & (x2,x ~) E < U I G > & (Xl,X 2) E p(d)

in

X

the

implies

(x{,x~) E p(d) •

UIG

An

a__bstract

is

a monokit.

A pregrame then

~

equivalence

is

a

game

(a ~ame

It (3)

is

is

surjection

classes

for

obvious

short) that

n__on-degenerate and

~-1({a}),

yields a CA.

is

every if

a pregame game

X ~ ~.

a partition

(3)

is If of

such

a regular it

the

is set

that pregame.

the

case,

Y

into

314

Every The

non-degenerate

elements

of

The e l e m e n t s

of

Active lose

players

in t h e

that

y

D

are passive

result

The e l e m e n t s

their

X

means

are

situation

x.

situation

The

players.

condition

(x,x') C p(d)

situation

x

The

over

notion

The

the

F =

where

and

(Ya) a C A

strategies subset

of

A

of

p : D

a.

cases

sense

the p a y o f f Every

and

~acAYa

that various

o f this

k(a,x)

game

and

have

of sets.

that each by

function;

player

d

the

prefers

the

equivalent

to

[6].

notion.

If,

that

A x X

>H(a,x)

noncooperative

and O.Morgenstern

is the

to e a c h

a-th

every

(3) o n e m a y

associate

game.

Y

game

(a g a m e

the e l e m e n t s H(a,x)

as an a b s t r a c t union

of is

game

of the sets

the c o r r e s p o n d i n g

index

x.

the c__anonical p r e s e n t a t i o n

precisely,

the V o r o b ' e v

that

x).

of the e l e m e n t

game has More

X =~acAYa such

(the n u m b e r

is the d i s j o i n t in

coordinate

abstract

f o r m of a V o r o b ' e v

Y y

[3]),

function

situation

(3), w h e r e

on

are

> H ( a , x ' ) ],

(4) c a n be p r e s e n t e d

form

of g a m e s A = D,

a in the

assigns

types

defined

of the p l a y e r

~

as above,

is a n o n - e m p t y

in p a r t i c u l a r ,

is the p a y o f f

game

X

H

(4) is a c l a s s i c a l

H

same meaning

is the s e t o f a l l

.

important

function

Neumann

the Ya

Vorob'ev

Conversely,

game

means

strategies

is e s s e n t i a l l y

it is a s s u m e d

product

[6]

of J . v o n

in the

(a E A ) ,

in t h e

here

. P o w ( X x X)

is a r e a l - v a l u e d

are players,

Ya

a in t h e

is the p r e f e r e n c e

by N.N.Vorob'ev

family

Moreover,

in

the V o r o b ' e v

G = @(F)

o f the

a.

means

x'

V a C A V x , x , C X [ (x,x ') 6 p(a)

Category Theory: Applications to Algebra, Logic and Topology. Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981

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