Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann,...
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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Forschungsinstitut for Mathematik, ETH, Z0rich 9Adviser: K. Chandrasekharan
24 Joachim Lambek McGill University, Montreal Forschungsinstitut for Mathematik, ETH, Z(Jrich
Completions of Categories Seminar lectures given 1966 in ZOrich
1966
Springer-Verlag. Berlin. Heidelberg. New York
All rights, especiallythat of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomec.hanlcal means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. O by Springer-Verlag Ber]/n 9Heldelber8 1966 Library of Congress Catalog Card Numbex 66-29802. PHnted in Germmy. T/de No. 7M4~
Acknowledgement
These notes contain at the Mathematical 1966,
an embryonic
course
at McGill
The author
Research
is indebted
stimulation
Institute
lectures
of the E.T.H.
given
in F e b r u a r y
in a graduate
1965. to McGill
to the National
for a Senior Research its hospitality,
of seminar
version having been presented
in spring
leave of absence,
their
an account
Fellowship,
for a generous
Research Council to the E.T.H.
and to Bill Lawvere and criticism.
University
of Canada
in ZUrich
and F r i e d r i c h
Ulmer
for for
Contents
Introduction
. . . . . . . . . . . . . . . . . . .
2
Terminology
. . . . . . . . . . . . . . . . . . .
6
G e n e r a t i n g and sup-dense
subcategories
Limit p r e s e r v i n g f u n c t o r s A sup-complete
sup-dense,
The c o m p l e t i o n w h e n The r e l a t i o n s h i p completeness Theorems w i t h o u t
~
. . . . . . . . . . . . sup-preserving
is not small
between d i f f e r e n t
10 17
e x t e n s i o n 24
.......
27
forms of
. . . . . . . . . . . . . . . . .
35
properness
42
conditions
Completions
of groups
Completions
of c a t e g o r i e s
of algebras
Completions
of c a t e g o r i e s
of m o d u l e s
References
......
......
. . . . . . . . . . . . . . ......
51 55
.......
58
. . . . . . . . . . . . . . .
69
-
O. Introduction. and inverse
limits of
The derivative preserving"
-
We shall call the generalized direct Kan
"supremum"
"sup-complete",
and "infimum"
"sup-dense",
respectively.
and "sup-
then also have fairly obvious meanings,
be made precise their duals
terms
2
which will
in the text. One can distinguish these terms from
"inf-complete"
from "right" or "property"
etc., without being able to tell "left" from "co-property".
Can every small category
A
be embedded as a (full)
dense subcategory into a sup-complete
category
also noted by others,
to be the category of all
functors from
-A~ ,
is yes: Take ~'
the opposite category of
A'
~,
?
sup-
to
The answer,
Ens,
the
category of sets. U n f o r t u n a t e l y the embedding does not in general preserve However,
consider
functors from
A._ O
instead the category
~'' of all inf-preserving
to Ens. The embedding of
is sup-dense and sup-preserving. it is an open problem whether
Moreover
A
~''
into
it is also sup-complete.
is sup- and inf-complete with a sup-dense,
of
A__~ ~'''.
To wit,
A.A'' which are subobjects
It is an open problem whether category
A''''
~''
is inf-complete;
Luckily there does exist at least one category
embedding
sups.
let in
A'''
which
sup-preserving
A_A''' consist of all objects ~''
of products of objects in
A~
there exists a sup- and inf-complete
with a sup- and inf-dense embedding
A ~ A''''
in analogy to the Dedekind completion of an ordered set.
-
3 -
Now let us drop the assumption still define all
A--
functors
this means
and
T :
~ Ens
in
D,
is still
sup-dense
A~
A--'
A'' A
is complete
is still
to
A in
A__' and
sup-preserving.
Ens
Theorem.
is known
equivalent
hand,
which
appears
(This result
While
A'
from some
such
y ~ T(D),
x = T(f) (y).
is sup-complete,
All proper
completeness"
inf-preserving
on
Adjoint
implies
is also a symmetric
theorem
of
functors
A.A are equivalent
A__ is
A_~
Ens,
to the corresponding
category.
to be slightly more
Adjoint
general
Functor
conditions
Theorem
than any in the literature.
to show the sup-completeness
We also give new sufficient
Functor
completeness
of certain
a form of the Special
imply sup-completeness.
to a general
There
to the
also announced
form of sup-completeness
-A ~ , the opposite
is required
relate
inf-completeness
This fact,
representation
to the r e p r e s e n t a b i l i t y
We obtain
A.
and the embedding
On the one hand,
that a m o d i f i e d
on
of
A'',
completeness.
and that both conditions conditions
comes
D
to Isbell,
such that
to be equivalent
On the other
asserts
According
subcategory
sense:
a kind of sup-completeness. which
one restricts
i.e.,
"representation
representation
by Benabou,
One m a y
are representable.
forms of completeness?
implies
be small.
provided
A,
f : A ~ D, in
A.
"proper".
a small
in a different
How does this older
to be
x ~ T(A),
via some'map
functors
as before,
that there exists
that every element D
A--'
that
of
~'''
for inf-completeness
to
above.)
-
4 -
To illustrate completions of small categories, sider the example in which
A
is a group
is the category of all permutational In another application,
we let
an equationally defined category operations.
If
A
many generators, of
~
.
A__'' = A__' of
G~
be a subcategory of
of algebras with finitary
contains a free algebra with sufficiently then
A__I' is equivalent to a
A__* consists of all algebras
preserves infs.
Then
representations A
~
G.
we first con-
When
~
subcategory
C such that
A__ w
[ -, C] : _A~ ~ Ens
is the category of all R-modules,
this
result was first obtained by Ulmer. Finally,
when
C
is the category of all
fairly generous conditions on C
such that every
Aj
A__* consists of all
nonzero submodule of
nonzero factor module in
R-modules,
C
under
R-modules
has a
A__. Prior to showing this, we make a
general study of certain pairs of classes of
R-modules,
as
exemplified by the following pairs of classes of Abelian groups: torsion,
torsion-free;
divisible,
reduced.
It will be assumed that the reader is familiar with what is common to the standard expositions of Category Theory [MacLane,
Freyd, Mitchell],
concepts:
category,
and epimorphism, pushout,
functor,
natural transformation,
monomorphism
subobject and quotient object, pullback and
Yoneda's Lemma
categories,
in particular with the foll~ving
[see MacLane,
adjoint functors,
p.54],
representable
equival~nce of functors.
-
Subcategory
5
-
will always mean full subcategory, e m b e d d i n g will
always m e a n a full and faithful
functor.
Some other well-known
concepts will be redefined in Section I, to allow for some idiosyncracy in terminology. I have attempted to make these notes readable, risk of including some so-called in the literature,
For proofs
the reader is sent to the recent book
b y Mitchell whenever possible. however,
"folk-theorems".
at the
For some important results
the papers by Isbell m u s t be consulted.
-
I_o T e r m i n o l o g y . A,
B
It is u n d e r s t o o d
of a c a t e g o r y
of m a p s
a : A ~ B.
of sets,
called
if the c l a s s
"A",
a.
[A, B]
for e v e r y p a i r of o b j e c t s a set
is i t s e l f
Hom
(A, B) = [A, B]
the o b j e c t
of the c a t e g o r y
this m a y be taken (1962)].
of o b j e c t s we do not
to be any of the u n i v e r s e s
The c a t e g o r y
is a set,
i.e.,
"category",
we
shall
consider
"object"
and
categories
A
A
is c a l l e d
an o b j e c t
find it c o n v e n i e n t
for
Functors
that
is g i v e n
"a"
Frequently maps
there
[see G a b r i e l
Regrettably, "A",
A_
Ens;
of G r o t h e n d i e c k small
6 -
to use
"map"
of Ens.
the
consistently.
with objects
w i l l be d e n o t e d b y c a p i t a l
styles
a
and
R o m a n or G r e e k
letters. A diagram
is the same thing
this t e r m i n o l o g y particular, dia@~am
is used,
I is c a l l e d
w i t h e a c h o b j e c t A of
A I : I ~_A,
defined
i
An U p p e r b o u n d of an o b j e c t (Of course,
A
or s u p r e m u m exists
diagram-
and m a p
~
of
(A, u) F
i
of
of
index category.
we m a y a s s o c i a t e
In
the c o n s t a n t
of I.
F
F : I -~ A_A c o n s i s t s
transformation sufficient
a : A -~ A'
u
w i l l be c a l l e d
such that
situation
: F-~ A I.
to s p e c i f y
if for e v e r y u p p e r b o u n d
I. This
If
(t) = 1 A,
(A, u) of a d i a g r a m
it w o u l d h a v e b e e n
a unique map
all o b j e c t s
AI
and a n a t u r a l
The u p p e r b o u n d
~
the
F : I ~ A.
by
A I (i) = A,
for e a c h o b j e c t
as a f u n c t o r
u
alone.)
a least u p p e r b o u n d
(A', u')
of
F
there
au (i) = u' (i) for
is i l l u s t r a t e d
by a commutative
-
A
.....
7
-
~- . . . . .
u(i)
~ A'
' (i)
r(i) We w r i t e Actually,
sup
F =
the o b j e c t
We shall use
(A, u), A
I A. Let
and o n l y if [F, A~] A', w h e r e A'
~ [A, A']
one m a y a l s o F, w r i t t e n
The
is the same
The or
A
over
of
inf
suggested
by various
by Amitsur.
a directed
free g r o u p s Other I
cokernel a, b
are c l a s s i c a l special
~i
F (i)
as s p e c i a l
cases.
isomorphism
in
of sup
cok
lower b o u n d : A I -~ F.
"co-limit",
direct
as a s p e c i a l limit~
k i n d of sup, every
but only torsion-
of free ones.
are th~ sum
"discrete" (a, b)
kar
limit
For example,
groups,
limits F
u
limit",
the c l a s s i c a l
direct
the k e r n e l '
if
The p r e s e n t ' t e r m i n o l o g y
[i~iF(i),
category,
of a p a i r
and the pull.back. D u a l l y ,
~I
F = A
A._ ~
"direct
free A b e l i a n
(or " c o - e q u a l i z e r " )
sup
(A, u), w h e r e
direct
is j u s t a set or small
.- A ~ A',
product
cases
isomorphism.
the g r e a t e s t
set I m a y be v i e w e d
is a sup of
case
F =
While
Abelian
F = A.
A.
authors.
is a c l a s s i c a l
group
. Then
as the sup in
not e v e r y sup
sup
Kan:
is a n a t u r a l
s u p r e m u m has b e e n c a l l e d
"right root"
was
of
introduce
or i n f i m u m of inf in
result
loosely
o n l y up to
F 9 I -~ A
is a n y o b j e c t
Dually,
sometimes
is u n i q u e
the f o l l o w i n g
PROPOSITION
and
(a '
in
the
of m a p s
inf
F
b)
and the p u s h o u t '
has
the
-
We shall every
call
diagram
following
with
if and o n l y
if e v e r y
sup-
into a more category
and
general
of all
A__ is small.
functors
PROPOSITION and
A
I
is small,
Proof
context.
The m a p s C.
If
C
then
~,
~]
diagram
F - I ~
objects
A
Let
F' (A) = (F(A),
sup
of
[A_A, C]. A
F (i) (A) ~ F(A). functor v
and
: F ~ GI
Put
One e a s i l y
: F ~ FI
is a n o t h e r
: F(A)
natural
is to say,
~ G(A)
such
fu(i)
= v(i) .
functors put
be
that
from
this
[_AA, ~]
assumed
that
(inf-complete) (inf-complete).
and c o n s i d e r
any
and w r i t e
for all
: I-~ C
is a functor.
u(i) (A) = u'(A) (i)
that
F
9 A-~ C
transformation.
transformation,
there
for the
= F(i)(A),
that
to Ens
transformations.
F' (A)
verifies
A
Moreover,
sup-complete
F' (A)(i)
I. Then
(kernel).
last o b s e r v a t i o n
, it b e i n g
small
and v e r i f y Then
~
and e v e r y
inf-complete.
are n a t u r a l
a natural
transformation.
and
is also
u'(A)),
v' (A) (i) = v(i) (A)
a natural f(A)
u
of
by
(inf-complete)
(product)
is s u p - c o m p l e t e
Define
and i
sup-
__A to
I
The
discovered
a cokernel
We shall w r i t e
[~, ~]
Let
has
We shall
from
of
(sketched).
has a sum
of all
inf-complete.
(inf).
is s u p - c o m p l e t e
same o b j e c t s
the c a t e g o r y
has a sup
if
and Maranda:
A category
the
(inf-complete)
to h a v e b e e n
that Ens is b o t h
_AA is small,
is b o t h
seems
set of o b j e c t s
of m a p s b e t w e e n It is clear
suP-complete
and Hilton,
I B~
-
index c a t e g o r y
result
Eckmann
PROPOSITION
if
small
fundamental
Grottendieck,
pair
a category
8
exists
is a
Suppose
where
v' (A)
:
G : ~
~.
: F' (A) ~ G(A)
a unique
map
f(A)u ~ (A) (i) = v" (A) (i) , that
Finally,
one v e r i f i e s
that
f 9 F -~ G
is
-
is a natural transformation.
9 -
It follows that sup F = (F, u).
This proof is illuminated by the isomorphism of categories
[ A , [ I , C]] = [IXA, C] ~ [AXI, C] ~- [I, [A__.,C]]. We have presented it here, because the construction of the sup in [A, C]
will be of importance
later.
-
~.
Generating
With
any
9nd s u p - d e n s e
functor
category
~
T
: A ~ Ens
we
shall
assume
A__ to d i s j o i n t
sets.
This
b u t even
if
construct we m i g h t
s u b c a t e @ 0 r i e ~.
and a d i a g r a m
simplicity,
T
put
we
that
which
[[A, -],
and
A ~ A'
hence
T'(A)
does
not m e e t
the
in the usual sets
T(A),
where
if also
~ =
such
set,
the e l e m e n t The c o n d i t i o n
x'~
distinct
that
T'(A').
of
objects
of
satisfied;
it. For example,
[~, Ens],
implies
a
it is e a s y to
does have
then
T' ~ T,
[A, -] ~ [A', -],
(The r e a d e r
will
it is a s s u m e d
that
Let
A
are
of
T(A')
recall
that
x' = T(a) (x). a'),
we put
is d e f i n e d
of the d i s j o i n t
triples
are o b j e c t s
x r T(A),
~ =
of
X T,
We w r i t e ~'~ =
FT(~)
and
a)
a : A ~ A'
~ : x -~ x';
(x, x''
by stipulating
and
(x, x',
I
that
a'a)
9
r T (x)
is the a such
a) {O} be a typical
according x ~ T(A)
to L a w v e r e the m a p
satisfied
b y the c o m m u t a t i v e
is the u n i o n
XT
(x', x'',
: ~-~
(x, x'
Remark 9 element
of
such that
~' = FT
A
and
XT
The d i a g r a m
that
of o b j e c t s
x 9 T(A)
is the
the sake
is f r e q u e n t l y
of a c a t e g o r y
A in _AA. The maps
is a m a p of and,
definition
canonically
[A, B] are all disjoint.)
The class sets
sends
T] in
Lemma,
that
associate
this property,
by Yoneda's that
T
assumption
T' = T
T' (A) =
shall
P T : X T ~ A__. For
does not have
a functor
I O -
by
diagram:
~
one-element (1964)).
We m a y
: {O} -~ T(A)
(x, x',
a)
in
set
associate
such may
(or the one-
that
with
R(O)
= x.
then be r e p r e s e n t e d
-
T(m
1 1 -
T(A')
-
T(a) We
shall u l t i m a t e l y
inf of _A
FT . T h i s m u s t be m a d e
b u t o n l y of
being
assumed
a canonical We
shall
[_AA, Ens] that
HoF T
or, b e t t e r is small.
case
FT
as a d i a g r a m HO : A ~
result H~
Let
F
/%
T~
the d u a l
:
~ AA .
XT~
says t h a t
A in
G
T h i s m a y a l s o be e x p r e s s e d G'
: B ~
[A_A O, Ens]
In the s p e c i a l calls
A
is d e f i n e d
a ~eneratin@ dually.
T
A~ ~
:
B
section,
FT Ens,
in
to v i e w this embedding
the a b o v e
is the sup
and c o n s i d e r
of the
a functor
if for a n y p a i T
in B there e x i s t s
by aaying
G
T
: G(A) ~ B
defined by
case when
concerning
the c a n o n i c a l
~enerates
b
= [A, -]
in the p r e s e n t
convenient
form:
b I, b 2 : B ~ B'
A_A and a m a p
H(A)
= [ -, A]. T h e n
takes
of d i s t i n c t m a p s object
H~
it
there does e x i s t
where
a functor
be a small c a t e g o r y
: A ~ BB . One
result
It is o f t e n
where
[A_A, Ens] ~
of
However,
consider
is the
is in fact the inf of the
.- X T o ~ __A- C o n s i d e r
[_AA O, Ens],
mentioned diagram
p o
T
a weaker
: X T ~ ~A
still,
T
is not an o b j e c t
T
Ens] O
[~, Ens] O.
w e shall
as
sense,
Fortunately,
H : A_A~ ~
shall be c o n t e n t w i t h
which
in some
precise,
in [~, Ens] O,
: XT ~
Sometimes
G
~
embedding
see that
diagram we
s h o w that,
such
that
an b i b ~ b2b.
t h a t the functor
G'(B)
= [G -,
B]
is faithful.
is the i n c l u s i o n
of
A
subcategor7
of B__t. The t e r m
in
B , one
"cogenerate"
-
PROPOSITION G
generates
2.1.
__B if e v e r y
o f a sup o f s o m e holds
when
First,
form.
Suppose
p
B.
blpu(i)
generated
B_ ~
Conversely,
functor
X
X
the
= x G(a).
for
1 .- G
l(x) o F
Now p
: B* ~
bl,
let
in
Bx
such
B.
Then object
The converse
F(~)
X,
B'
[A,
sake
= a. ~
B,
and hence
b I = b2.
B
of objects of
= [G(a),B](x) x
: G(A)
[G(A),B]
~
B
are
~ [G(A),B], verified
it is e a s i l y
that
transformation.
(B*,u),
= b2P
that
x
the
G
the m a p s
x'
sets
ifrom
category
class
that
A.
is a s s o c i a t e d
and
any object and
then
pu (x) = x. blP
A_j
in
a l s o ithat
small
The
the
B, A
Thusl
and
the
such
that
For
is a n a t u r a l =
A
is the
~
there
B
such
a)
a
It f o l l o w s
B,
in
the
a n d an e p i m o r p h i s m
: G(A)
X ~ A. A
has
bI = b2.
hence
B],
(x, x',
: G(F(x))
b I.
in
-~ _B,
B
then we have
generates B
of
(B*, u),
in
F = FT ~
F(x)
that
i
G
: ZA
of all
sup(GoF)
b2 : B ~
all x
that
B]
B',
= b2P , hence
simplicity's
= x
-~
B
blP
~ =
and
object
for all
any object
Moreover,
all disjoint), put
: A ~
F : I ~ A.
(GoF) =
= b2b
the diagram
triples
(we a s s u m e ,
we
With
is the u n i o n
are
G
is a q u o t i e n t
where
for all
that
T = [G -,
and
sup
blb
assume
is s u p - c o m p l e t e o
of
of
b 1, b 2 : B ~
= b2Pu(i)
of sup
X = ~o
small,
that every
with
Suppose
definition
the
Go~,
assume
F : I ~ A,
Then
be
object
diagram
diagram - B* ~
A
_BB is s u p - c o m p l e t e .
Proof. indicated
Let
12-
there
Moreover would
Thus
yield B
exists p
a unique
is epi, blx
= b2x
is a q u o t i e n t
map
since for object
=
-
of sup
(GoF),
Again
and
consider
Following
Isbell
i.e.,
adequate
B_~
sup-dense
we call
H~
A ~
:
embedding
following
has
B
A
T
Let
B_B.
if
G
We shall
is
is left call
G
When
G
is the
inclusion
or s u p - d e n s e by Isbell,
of
_BB,
_BB,
Right
adequate
For example,
is right
the
adequate.
The
b y Ulmer:
is small, then
: A__~ ~ Ens
o
in
the c a n o n i c a l
dually.
[_AA, Ens]
A
subcategory
is left adequate.
H : _AA ~
of
B
is sup-complete,
G
is sup-dense.
and
In p a r t i c u l a r ,
is the sup of r e p r e s e n t a b l e
functors
this r e s u l t
holds
and will
even w i t h o u t be p r o v e d
the a s s u m p t i o n
in g r e a t e r
generality
5.
Proof.
In v i e w of P r o p o s i t i o n
F 9 I -~_AA,
: B* -~ B
Thus,
Ens]
is the sup of some d i a g r a m
are d e f i n e d
is s u p - c o m p l e t e
in S e c t i o n
p
[A__~
_BB
in A.
Actually,
we have
:
for
B
[A o , Ens]
A
A__ is small.
in
is left adequate,
functor
B
If
G'
full.
also b e e n o b s e r v e d
COROLLARY.
that
and
where
left a d e q u a t e
generates
object
functors
canonical
[-, A],
faithful G
G
functor
As had b e e n n o t e d
and i n f - d e n s e
every
we call
_AA a left a d e q u a t e
embedding
: A~
G : A ~__B
F : I ~ A.
respectively..
G
a functor
then
if e v e r y
G ~ F, w h e r e
is complete.
associated
an embedding, for
the p r o o f
(1960),
if the c a n o n i c a l l y
1 3 -
such t(A)
t(A) (x) = u(x).
sup (GoF)
that pu(x)
: [G(A),
=
2.1
(B*, u) ,
, or rather
its proof,
and an e p i m o r p h i s m
= x.
B] -~ [G(A),
It is e a s i l y
B *]
verified
be d e f i n e d that this
by
is n a t u r a l
in
A,
-
hence
we h a v e Now
the
functor
the m a p p i n g
hence
Therefore that
a natural
there
in
x
Therefore sup F =
X
B],
[G -,
= B ~ B*
(see the p r o o f
(B, I),
is a s u p - d e n s e dually
and so
pbp = p, h e n c e
As we have
the
where seen,
[G -,
to be
B*].
full,
is onto.
B*]]
[ G -,
such that
b]
= t,
embedding
functor
[A_AO, Ens] H~
the does
rationals
not
in g e n e r a l
due
of integers
the same d i r e c t e d
in the c a t e g o r y
functor
of
direct
We saw that a n y it a functor
An e a s y c o m p u t a t i o n
: _BB~
shows
G
Thus
is complete.
H O 9 A.A~ [A_A O, Ens] category,
it not While
hence
to call
for the many
The A b e l i a n
with
embedding
fact
examples
9 and I0, we m e n t i o n group
of
isomorphic
denominator
has d i r e c t
limit
n),
zero
groups. : A - ~ _BB has c a n o n i c a l l y
[AAO, Ens],
that
is epi.
limit of s u b g r o u p s
set of g r o u p s
functor
G'
(1966)-
(all f r a c t i o n s
of free A b e l i a n
sups.
in S e c t i o n s
to U l m e r
of sup.
be t e m p t e d
Aj were
preserve
Thus
is an i n f - d e n s e
One w o u l d
systematically
example
p
and the p r o o f
into a s u p - c o m p l e t e
category.
2.1).
by definition
H 9 A__~ [A, Ens] O
is a c l a s s i c a l
to the group
= x,
"sup-completion"
be d i s c u s s e d
n o w a simple
bp = I,
also bp = I, since
l(x)
= bx,
of P r o p o s i t i o n
the c a n o n i c a l
into an i n f - c o m p l e t e
with
b
B] ~
is a s s u m e d
= t(A) (x) = [G(A),b](x)
bpu (x) = bx = u (x),
but
[A_O, Ens]
[[G -,
a map
t : [G -,
is to say
for any
will
: B ~
[B, B*] ~
u(x)
that
transformation
G'
exists
14-
where
G'o G "--Ro.
G' (B)(A)
associated
= [G(A),B].
-
In particular, such that
H'o H ~ H ~
[~, Ens] ~ to verify
there
that
H~
H O'
maps
there
o
composite
T* ~ T*
not enough
to establish
and
of all U +.
out,
+
that
"reflexive" T *+ ~
T
In this way,
and
all
[.AAO, Ens]
and
is rather
small.
When
elements,
Isbell
(1964)
three objects, G~
[A, Ens] ~
the regular
and two other trivial
(see Section
the c a t e g o r y
such that
G
functors,
with more
hence
> 4-
A
such U +* m U. of
than two
representation
[~,
has
just
of
is far from complete
,.-~--
T(A')
with
Hence there exists a unique T(a')
t = R
t ~ T(P).
and
Since
of
B, it follows that any representative
of
B
is a dominating
set for
T.
T(k') P
t = 9. .
is a subobject
set of subobjects
-
Before ideas if
stating
of Isbell
m = m'e'
He p r o v e d
map
A__ has f
He c a l k e d
e' epi
we s u m m a r i z e
a monomorphism
implies
that
e'
some m
extremal
is an i s o m o r p h i s m .
this:
PROPOSITION of
our n e x t result,
(1964).
and
4 5 -
7A.
If
A__ is i n f - c o m p l e t e
a representative
of
A
has
and e v e r y o b j e c t
set of subobjects,
a canonical
decomposition
then e v e r y
fe
is epi
e
and
decomposition product
where
f = fm f
--
f m
is an e x t r e m a l
is u n i q u e
of e x t r e m a l
monomorphism,
up to isomorphism.
monomorphisms
and this
Moreover,
is again
#
the
an e x t r e m a l
monomorphism. From
this we deduce
LEMMA complete objects. e epi,
7.1.
(Diagonal
and e v e r y If
then
the
there
Lemma.)
object
mg = he,
following:
has
m
exists
Assume
that
A
a representative
an e x t r e m a l a unique
g
d
is inf-
set of sub-
monomorphism such
that
and
m d = h.
> J
e
d 4
~
J f / f j
h Proof L
decompositions.
canonical
g = gmg e
Let
Then
x
and
h = hmh e
(mg m) ge = h m (hee)
decompositions
an i s o m o r p h i s m
'>
such
of that
fo
be the c a n o n i c a l
= f, say,
By u n i q u e n e s s ,
Xge = hee
and
are
there
two
exists
h m X = mg m.
-
Take d
d = -mU x
-1
he,
then
46
-
Since
md = h.
m
is mono,
is unique with this property. ge
.>
L
~[
gm
I
->
x I I
m
! !
>
m
h
PROPOSITION
is also sup-complete CASE I. consisting of
A
O
A
Let
CASE 2.
A_ be inf-complete.
_A
A
contains
two cases:
subcategory
Ao, and arbitrary
sums
a generator
A. Moreover,
Ao, and arbitrary
every object in
A
- -
has representative Proof.
sets of subobjects
For any set
X, let
the direct sum of copies of x ~ X. Then
G : Ens ~ A
right adjoint
=
A__
A.
exist in
F = [A O, -]
Then
in either of the following
contains a left adequate
O
[X, F(A)]
m
of a single object
exist in
sums of
> h
e
7.2.
%
= ~xeX
= [ x~X Ao
denote
one for each element
is a functor, which has as a
the so-called
" A___~ Ens.
Ao,
G(X)
and quotient objects.
for@etful
functor
Indeed,
x], F(A
= TIxex [
A] ~
e
O'
=
-
It follows
47
-
that there e x i s t c a n o n i c a l
e : GoF -~ 1
and
m
: I -~ FoG
natural
transformations
with well-known
universal
properties . Consider
any d i a g r a m
r : I-~ A, w he re
In v i e w of the c o r o l l a r y show
that
F
subcategory exist x(i)
D
is proper.
D
in
of
= (G(V),
(FeF)
=
i
Since
map
f : G(V) -~ A
in
such that
e(A)
: G(F(A) ) -~ A
(b)
F(e(A)):
let
x r FCG(A))
universal
sups
also sup
(see
(GoFoF)
: G ( F ( F ( i ) ) ) -~ A
is
a unique
= x(i)
e(F(i)).
facts:
is epi.
of the fact that
A O is a ge ne ra to r.
(a), bu t is shown directly.
= [Ao, A] = [GC{O}),
p r o p e r t y of
preserves
f G(v(i))
-~ F(A)
(a) is an e a s y c o n s e q u e n c e from
such that
is epi.
F(G(F(A)}~
(b) is not d e d u c e d
G
i, there exists
We shall use the f o l l o w i ng (a)
n : D -~ A
3A), he nc e
x(i)e(F(i))
e a s i l y seen to be n a t u r a l
x : F -~ A I , there
I.
(V, v). No w
to P r o p o s i t i o n
e(v)).
and
in
to find a sm al l
given
y : F ~ D I,
is smal~.
6.2, we n ee d o n l y
Thus we w a n t
so that,
for all
sup
the c o r o l l a r y
A
D,
= n y(i) Let
to P r o p o s i t i o n
I
A].
e, there ex is ts a
Then,
unique
Indeed,
by the f : {O} -~ F(A)
such that X = e(A)e(f)
= [A O, e ( A ) ] ( G ( f ) )
= F(e(A))(G(f)).
-
48
-
G(f) G(F (A))
---G({O})
~
y
e(A)I, A
In view of the preliminary
spadework done by Isbell,
Case 2
will be a little easier to deal with than Case I. We shall therefore
consider
where
is an extremal monomorphism
m
Since
e(F(i))
it first.
is epi
the Diagonal Lemma y(i)
: F(i) ~ D
verified
that
quotient
(see
y(i)
objects
of
and
= x(i)o
in
i, h e n c e
to be a r e p r e s e n t a t i v e G(V).
> G (V) i e j, J
I P P
P D
y(i)/ ..:~ /
r(i)"
with
pf %
I I
m~.,~
J
x(i)
e
image
D,
is epi.
and obtain a unique map
m y(i)
is n a t u r a l
D
f = me
(a) above), we may apply
(Lemma 7.1),
such that
Thus we may take
Let
> A
It is e a s i l y y 9 F-+ D Iset of
- 49In Case
1
we shall consider
but after applying
the forgetful
the same square as above, functor
F.
!4
F(f)
//m (W)
s (i) /
i ~
m
////~i)
F(m~) ~ ~
~ F (A)
F(x(i)) Write
F(f)
with image
= me, S.
where
Since
m
is mono and
e
F(e(F(i) )) is epi
(see
there exists a unique mapping
s (i)
It m a y be verified
is natural
Now c o n s i d e r By the universal m'
: G(W) ~ A
= re(W) s(i),
set
m'y(i) D
s(i)
the canonical map property of
such that then
will be complete that
that
such that
m(W)
= m.
= m s(i)
if we can assert that
= x(i).
in
(b) above), m s(i)
= F(x(i)).
i.
: W -~ F(G(W) ).
m, there exists a unique
F(m')m(W)
F(m')t(i)
is epi in Ens,
Write
t(i) =
= Fix(i)).
Our argument
t(i) = F(y(i))
such
For we m a y then take the dominating
to be the set of all
G(W),
where
W
is any
-
quotient valence
set of
FIG(V)),
5 0 -
defined b y an equi-
let us say,
relation.
To p r o v e more general
the a b o v e
assertion,
let us c o n s i d e r
the
situation:
F(B)
>
F(A)
F(C)
P (v)
It is a s s u m e d seek
w
: A-~ B
Since
CAo}
the m a p p i n g and onto.
u
s u c h that
F(u)x
x = F(w)
-+
[F(A),
it is onto,
u w = v.
2, see also
= F(v)
and
subcategory F(B)]
Benabou
F(uw)
. We
u w = v. of
A_~
is o n e - o n e
there e x i s t s
Therefore
is one-one,
For case
and
is a left a d e q u a t e
x = F(w). F
: B-~ C
F : [A, B]
Because
s u c h that Because
that
w
: A ~ B
= F(u) x = F(v).
T h i s completes the proof.
(1965),
Th~or~me
5.
-
~. CQmpletions [A__ O, EnS]inf
when
of 9roups.
We wish to investigate
A_A is some known small category.
our first example we take one object,
5 1 -
A
to be a group
we may as well call it
(= elements of
given any two objects
r
[ik, ik+l]
for any or
and
I
I
connected
...
if,
in = j
'
k = O, 1, . ., n - 1,
[~+I'
and a
i, j ~ I, there exist objects
i ~ = i, i I ' i2,
such that,
has
G) are all isomorphisms.
F : I ~ A_. We call
(*)
G :A
G, and the maps of
Now consider any small index category diagram
In
~]
one of
is nonempty.
What do the lower bounds of a connected diagram look like? Consider
Suppose
(G, s) is a lower bound of F.
two neighbouring
+IJ
or
case we have a map
F
indices
~'
~+I'
so that
I, % 1
is not empty. In the first
Lk : ~
-~ ik+ I.
By naturality, -i
F(tk)
s(ik+ 1) = s ( ~ ) ,
hence
In the second case we m a y take s(ik+ I) = F(t k) s ( ~ ) . determined by
s(~).
s(ik+l)
tk " ~ + 1
Thus in either Applying
= F(L k) "~ ~
case
s(i k)and
s(ik+ I)
this to the sequence
is (*), we
obtain
(**)
s(j)
=
r(t
n-
1 )+-1 . . .
r ( ~ l )+1 r ( , o )~1
s(i)
= g s(i).
-
PROPOSITION
8.1.
Let
A__ be a g r o u p
is a lower b o u n d
of a c o n n e c t e d
inf
If
F =
(G, s).
connected
diagram
Proof. Given
two
(*), we of
G
of
i
have
-1
~ G.
t(j)
= s(j)h,
(We r e g a r d
F : I ~ ~,
F
is a c o n n e c t e d
j
which
are c o n n e c t e d
= g s(i),
where
then a dis-
g
t(j)
(G, s) j
fixed,
diagram. b y a sequence
is the e l e m e n t = s(j)h,
where
is the o n l y e l e m e n t
h
hence
as
(G, s)
has no info
Clearly
i
If
than one element,
(**). T h e r e f o r e
t(i)
that F.
by
G.
that
and t(j)
determined
h = s(i) such
assume
indices
diagram
has m o r e
F : I ~ A
First
then
G
52-
is in fact
as a n y e l e m e n t
of
G
the inf of
I.)
g
s(j)[ G
~"
G
F(j) Next
consider
m a y assume 11 and
iI
Let I
of
I
is the u n i o n
such that
12
for all
h/
that
a disconnected
in
(G, s) G
as follows -
11
[il, and
be a n y
and define
in
[i2,
i I]
We
categories
are e m p t y
12 9
lower b o u n d a new
F z I ~ A~
of two n o n e m p t y and
i2]
i2
diagram
of
F.
lower b o u n d
Take
any e l e m e n t
(G, t)
of
F
-
t(i)
Clearly
This
= s(i)
if
i
is in 11
= s(i)h
if
i
is in 12
there
for all
i
is no e l e m e n t
in
I.
completes
then
inf
8.2.
If
F =
But then also
G
F =
Surely show
If
that
t(i)
is n o t the
= s(i) g
inf of
P.
G, e v e r y
has o n l y one e l e m e n t
inf
= (T(G),
(TOP)
li(i)
1
= 1
and
F : I -* G,
for all
i
TolI),_ as is e a s i l y
has m o r e
(G, s). Then
I
is connected,
(T (G), Tos)
functor
infs.
G
that
such
is a g r o u p
where
than one e l e m e n t
ToP,
any other
let
i
b y the
sequence
lower b o u n d
b e a fixed (*). Put
in v i e w of
j
f = TIs(i) -I) u(i)
of
T(P(to)u,
u(~+ 1) = T(P(Lk )-+1) u(i~),
+I)
of
any index,
(**),
= T(F(tn_l) -+1) ... by naturality
index,
(X, u)
u
(i)
.
: X
I.
verified.
and that
by Proposition
is a lower b o u n d o f
in
8.1.
we w i l l
it is the inf.
Again,
Now,
A
that
Consider
Then,
G
(G, s)
(G, 1i) ,
N o w assume inf
of
Therefore
: ~ ~ Ens p r e s e r v e s Proof.
g
the proof.
PROPOSITION
T
53-
To F. connected -~ T(G).
- 54-
for
k = O, I,
the above,
..., n - 1.
(F(G),
f
Fos)
this r e p e a t e d l y
to
we obtain
T~sCj))
Moreover,
Applying
f = uCj).
is clearly unique with is indeed
inf F,
this property.
and so
F preserves
Thus infs,
as to be shown.
f F (G)