Stone spaces

Stone spaces

P E T E R T. J O H N S T O N E Assistant Lecturer in Pure M athem atics, U niversity o f Cambridge

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U N IV E R S IT Y

PRESS

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Published by the Press Syndicate of the U niversity o f C am bridge The P itt Building, T rum p in g to n Street, C am bridge CB2 1RP 32 E ast 57th Street, N ew Y ork, NY 10022, USA 10 S tam ford R oad, O akleigh, M elbourne 3166, A ustralia © C a m b rid g e U niversity Press 1982 F irst published 1982 F irst paperback edition 1986 P rin ted in G reat B ritain at the U niversity Press, C am bridge

L ibrary of C ongress catalogue ca rd num ber 82-4506

British Library cataloguing in publication data Johnstone, Peter T. Stone sp a c e s—(C am bridge studies in advanced m athem atics; 3) 1. Topological spaces I. Title 514'.3 QA611 ISB N 0 521 23893 5 hard covers ISB N 0 521 33779 8 paperback

Contents

Preface Advice to the reader Introduction : Stone's Theorem in historical perspective

vii ix xii

I 1 2 3 4

Preliminaries Lattices Ideals and filters Some categorical concepts Free lattices N otes on chapter I

1 1 11 15 25 35

II 1 2 3 4

Introduction to locales Fram es and locales Sublocales and sites C oherent locales Stone spaces N otes on chapter II

39 39 48 62 69 76

III 1 2 3 4

Compact H ausdorff spaces C om pact regular locales M anes’ Theorem Gleason’s Theorem Vietoris locales N otes on chapter III

80 80 92 98 111 119

IV 1

Continuous real-valued functions Complete regularity and U rysohn’s Lem m a

123 123

V

vi 2 3 4

The Stone-Cech compactification C(X) and C*(X) Gelfand duality N otes on chapter IV

130 142 152 164

V 1 2 3 4

Representations of rings A crash course in sheaf theory The Pierce spectrum The Zariski spectrum O rdered rings and real rings N otes on chapter V

169 169 181 191 206 220

VI 1 2 3 4

Profiniteness and duality Ind-objects and pro-objects Profinite sets and algebras Stone-type dualities General concrete dualities N otes on chapter VI

224 224 233 246 253 267

V II 1 2 3 4

Continuous lattices C om pact topological (semi) lattices C ontinuous posets and lattices Lawson semilattices Locally com pact locales Notes on chapter VII

270 270 286 297 308 321

Bibliography Index o f categories Index o f other symbols Index o f definitions

324 364 366 368

Preface

It was in the summer of 1977 that it first occurred to me that there was no single text where one could obtain a balanced view of all the m athem atical consequences that have flowed from the Stone R epresentation Theorem, and that it would be useful to have such a book. At that time, however, I did relatively little to pursue this idea; the only thing th at I wrote down was a tentative list of chapter headings, which bore relatively little resem­ blance to the book which eventually emerged. I made a more serious start in the autum n of 1978, when I gave a P art III (graduate) course in Cam bridge entitled ‘Stone Spaces’; this covered most of the m aterial in chapters I-IV (and would have covered more, but for lack of time). I had the opportunity to recycle a good deal of the m aterial from chapters II and III in January 1979, as part of a course on ‘Internal and External Locales’ which I was invited to give at the Universite Catholique de Louvain in Belgium; but th at course went on to consider topos-theoretic applications of locales (written up in [Johnstone 1979]) which were never intended to form part of this book. The text of the first three chapters (except for section III 4) was w ritten up in the summer of 1979; in the autum n which followed, I gave a con­ tinuation of the ‘Stone Spaces’ course (to a subset of the original audience) which covered most of the m aterial in chapters VI and VII. The writing of the rem ainder of the text was largely done in the two succeeding summers: chapter IV and sections III 4 and V 1 in 1980, and the rest in 1981. After the text was completed, but before the typesetting began, I had the opportunity to polish up one or two points as a result of a further course of lectures during my sabbatical at M cGill University, M ontreal, in the w inter of 1982.

viii

Preface

In writing a book of this kind, one inevitably accum ulates m ore debts of gratitude than can be repaid in a short Preface. M y first debt is to the audiences of the lecture-courses m entioned above: particularly to my colleague M artin H yland and my student Andrew Pitts, whose unfailing enthusiasm for the project did a lot to keep me going; to Francis Borceux, who was responsible for inviting me to Louvain-la-N euve; and to M ichael Barr and M arta Bunge, who invited me to M ontreal. O ut of the many people who have contributed to my own education in the subjects covered in this book, it would be entirely invidious to select only three; but I shall do this by naming B ernhard Banaschewski, John Isbell and Andre Joyal. The influence of the first two can be gauged by the frequency with which their names appear in the N otes at the ends of chapters, and by the length of their entries in the Bibliography; JoyaPs contribution cannot be similarly m easured by w hat he himself has written, but his influence on my thinking about locales is nonetheless profound (see [Johnstone 1983]). Prelim inary copies of the typescript were circulated to a num ber of colleagues (including M ichael Fourm an, R udolf Hoffmann, D ana Scott, H arold Simmons and Myles Tierney), several of whom offered valuable com m ents and suggestions for im provem ent; in this context I must particularly thank Saunders M ac Lane for his expert advice on historical matters. (However, neither he nor any of the others m entioned should be held responsible for any errors which rem ain; they are mine alone.) Finally, and by no means least, I have to record th at my life has been enriched, since I began w orking on this project, by getting to know M arshall Stone personally. His courteous hospitality, and his keen interest in the present-day descendants of his fundam ental theorem s of the 1930s, have m eant a great deal to me. It remains only for me to thank D avid T ranah and his colleagues at C am bridge University Press for their efficiency in the production of the book, for their willing acceptance of all my unreasonable dem ands in m atters of style, and for their meticulousness in keeping me in touch with all stages of the production process - despite the best efforts of the C anadian Post Office to frustrate them during my stay in M ontreal. Cambridge, July 1982

P.T.J.

A dvice to the reader

Like a great many research-level books in mathem atics, this one is an uneasy compromise between a textbook for the student and a reference work for the specialist. The specialist will presum ably need no help in finding w hat he wants from the book (assuming it’s here at all); so these rem arks are primarily addressed to the student, or to the lecturer who might be considering using the book as the basis for a graduate course. First, prerequisites: the reader is presum ed to know about as much algebra and general topology as he might have been expected to pick up in a British undergraduate course. In particular, he is presum ed (in chapters IV and V, at least) to have some familiarity with com m utative rings; but on the other hand, the treatm ent of lattices is entirely selfcontained. (However, the book should not be regarded as a textbook on lattice theory - it misses out far too m any im portant concepts, in particular th at of modularity.) The treatm ent of categories (which are used freely throughout the book) is not self-contained; a student who has not met categories before will have to do some background reading to flesh out the bare bones in section I 3. Nevertheless, it would be possible for a course based on the book to proceed simultaneously with a first course in category theory; I should hope that the two would reinforce each other to a large extent. Similar rem arks apply to sheaf theory, which is used only in chapter V ; the first section of this chapter is not a self-contained intro­ duction to sheaves, but in conjunction with a first course in sheaf theory it should be sufficient to unlock the rest of the chapter. The num bering system used is rather old-fashioned: each chapter is divided into four sections (a sheer coincidence), and each section is divided into a num ber (between 5 and 17, but usually around 10) o f ‘paragraphs’, each of which can be regarded as the w orking-out of a particular idea. ix

x

Advice to the reader

The theorems, lemmas, corollaries, etc. are not num bered; but since every paragraph contains at m ost one of each, they can be referred to by their paragraph numbers. Thus T heorem III 2.4’ means the unique theorem in paragraph 4 of section 2 of chapter III. (For references within a given chapter, the chapter num ber is omitted.) To try to cope with the conflicting dem ands of student and specialist, I have labelled certain paragraphs as ‘secondary5, particularly in the first three chapters. These contain m aterial which is included either for the sake of completeness o r because it is going to be needed in a later chapter, but which may be om itted at a first reading w ithout dam aging the con­ tinuity of the narrative. They are distinguished by being m arked with an obelus ( f ), and printed in slightly smaller type. In the last four chapters, there are a num ber of paragraphs of prim ary m aterial which depend on secondary paragraphs in the first three chapters; obviously, when one encounters one of these, the remedy is to go back and read the m aterial which was om itted earlier. (F or example, on reaching paragraph IV 2.4 it will be necessary for anyone who has previously om itted paragraphs II 3.5-3.7 to go back and read them.) There is only one com parable instance within the first three chapters, where Lem ma II 2.8 is used in the proof of C orollary III 1.3; unfortunately it proved impossible to rearrange the m aterial to avoid this, but the earlier lem ma can easily be read out of context. The exercises are scattered throughout the text, instead of being segre­ gated at the ends of chapters. This is because they are really an integral p art of the text, and should be regarded as com pulsory for all readers - the result of an exercise is frequently used in a proof in the very next paragraph. F o r this reason, hints are given for the solution of all but the most routine ones. Because of the diverse nature of the m aterial covered, the logical depen­ dence relation between the chapters is m uch m ore fragmentary than is usually the case. The 'com pulsory core’ of the book consists of the In tro ­ duction and the first two chapters, which are prerequisites for all that follow; thereafter, chapter III is a prerequisite for chapters IV and VII, but th at is ab o u t all. (There are quite a num ber of cross-connections between chapters IV and V, and a lesser num ber between chapters VI and V II; but in each case it would be possible to read the later chapter w ithout having read the earlier one.) The Bibliography is quite extensive, but even so it does not claim to be a comprehensive listing of all the papers relevant to topics covered in the book. References to the Bibliography are by the name(s) of the author(s)

Advice to the reader

xi

and date of publication, enclosed in square brackets; except th a t when the author’s name occurs naturally in the sentence where the reference is made, only the date is given in brackets. W here the Bibliography lists more than one publication by a given au th o r in a given year, suffix letters are used to distinguish the second and subsequent ones. As far as possible, the dates given are those of first publication (except where specific refer­ ence is made to a second or later edition); but details of second editions, translations, etc. are given in the Bibliography itself where appropriate. The year 1984 is used as a conjectural publication date for forthcoming papers about which no more precise inform ation is known. This final paragraph is addressed prim arily to logicians. There is little that is overtly logical in this book, except in chapter V where a certain am ount of first-order logic is inescapable. This doesn’t imply th at I am uninterested in logic; on the contrary, I regard it as one of the m ost im por­ tant features of the theory of locales that it enables one to give construc­ tively valid proofs of many results whose counterparts in point-set topology are essentially non-constructive. However, I don’t see the need to clutter up a book about m athem atics with a lot of references to the logical framework within which one is doing the m athem atics: if an argum ent is constructively valid (and where possible, my argum ents usually are constructively valid), a professional logician will not need to be told this, whereas the sort of hard-nosed 'w orking m athem atician’ who regards logic like a disease will not th an k you for telling him anyway. (I hope th at he might, however, notice the fact th at a constructively valid proof of a given theorem is generally more elegant than one which relies heavily on the law of excluded m iddle; constructivity is alm ost as much a m atter of style as of logic.) O n the other hand, I have not been able to prevent a certain obsession with the axiom of choice from breaking through, particularly in the N otes on the first four chapters. W ithin the main text of the book, those theorems, lemmas, etc. whose proofs require (some form of) the axiom of choice are distinguished by being m arked with an asterisk; I hope that this will not prove distracting to those who don’t want to be bothered with such things.

Introduction Stone’s Theorem in historical perspective

This book is about a particular theorem - the Stone Representation Theorem for Boolean algebras - and some of the m athem atical conse­ quences which have developed from it in the last 45 years. Inevitably, the au thor of a book which sets out to chart the developm ent of a m athe­ m atical idea in this way is faced with the necessity of com prom ising between two approaches: the historical, in which one attem pts to follow each strand of the developm ent in m ore or less chronological order (but perhaps misses some of the interconnections between the various strands), and the logical or ‘genetic’ [M ac Lane 1980], in which one uses hindsight to take the most economical and painless route to the m ain results (but thereby loses some insight into why these results ever came to be seen as important). The particular com prom ise which I have adopted is to go fairly whole­ heartedly for a logical approach in the text itself (the route by which we shall eventually arrive at the proof of Stone’s Theorem in section II 4 will strike historically-minded readers as perverse, to say the least), but to begin the book with an Introduction which attem pts, first to set the R epresentation Theorem in the historical context in which Stone proved it, and then to indicate what those subsequent developments were, which led to the point at which the line of exposition I have adopted can be seen to be (as I believe it to be, anyway) an efficient and unifying way of covering a certain rather diverse body of m athem atical knowledge. (To reinforce the message of this Introduction, there are also sections of historical and bibliographic notes at the end of each chapter.) O u r historical survey begins with the birth of abstract algebra, which has recently been docum ented by Saunders M ac Lane in an adm irable essay [1981]. M ac Lane traces the first clear instance of an abstract/ axiom atic approach to algebra to a paper of Cayley [1854] on group xii

Introduction

xiii

theory. However, group theory was not in the forefront of the drive tow ards abstraction in algebra which occurred in the early years of this century; perhaps this was because Cayley’s representation theorem [1878], by showing that every abstract group was abstractly isom orphic to a ‘concrete’ group of substitutions ( = perm utations), removed the need for any abstract development of group theory until a much later date. If group theory is the oldest branch of abstract algebra, Boolean algebra has a good claim to be the second. O f course, Boole [1847, 1854] and Peirce [1880] were really only concerned with concrete algebras of pro p ­ ositions (or of classes), but W hitehead [1898] and H untington [1904] both took an abstract approach. However, there seems to have been little interest in non-Boolean lattices before 1930 (apart from the rem arkable papers of Dedekind [1897, 1900], which, however, were again concerned with concrete lattices - in this case lattices of ideals), and little development even of the Boolean theory beyond mere juggling with axioms. N ow although Cayley’s representation theorem may have delayed the developm ent of abstract group theory, it did at least stabilize the axioms of the subject by dem onstrating that they were indeed sufficient to capture ‘the algebra o f substitutions’. In Boolean algebra, there was a clear need for a similar representation theorem to show th at the axioms had cap­ tured ‘the algebra of classes’; but it was not immediately forthcoming. O f course, we should not expect such a theorem to say that every Boolean algebra is isom orphic to the algebra of all subsets of some set; for just as full perm utation groups have certain group-theoretic properties not shared by all groups (for example, if we exclude the group of order two, the property of having trivial centre), so there are lattice-theoretic properties enjoyed by full power-set algebras but not by all Boolean algebras. Let us briefly consider two of these. In the algebra of all subsets of a set we have, in addition to the binary operations of union and intersection (which are represented by the lattice operations v and a ), the additional possibility of forming unions and intersections of infinite families of subsets. We say th at a lattice is complete if it has infinitary operations V, A corresponding to these set-theoretic ones; it is easy to give examples of Boolean algebras which are not com ­ plete. Again, in the full power-set P X of a set X, the singleton subsets {x}, x e X , play a special role: they are not equal to the least element 0 , but there is nothing strictly between them and 0 - equivalently, {x} can­ not be represented as a union of strictly smaller subsets. An element of a Boolean algebra with this property is called an atom ; the abundance of atom s in P X is expressed by the fact that, for every Y=h0, there exists

xiv

Introduction

an atom Z with Z ^ Y A Boolean algebra with this property is called atomic; again, it is easy to give examples of non-atom ic Boolean algebras. N ow let B be an abstract Boolean algebra, and let X denote the set of all atom s of B. We may define a m ap <j>:B-*PX by setting (/>(b) = {x e X |x ^ b } . It follows easily from the definition of an atom th at an atom x satisfies x ^ b v c if and only if either x ^ b or x ^ c ; from this we may deduce that <j>is a hom om orphism of Boolean algebras. M oreover, (/> is one-to-one if B is atomic, since if b=/=c then the symmetric difference b A c lies above some atom , which will be in ju st one of <j)(b) and <j)(c); and is surjective if B is complete, since then any subset Y of X is the image under <j) of its join in B. Thus we have proved T heorem A Boolean algebra is isom orphic to the algebra of all subsets of some set if and only if it is complete and atomic. This theorem was first proved by the logicians A. Lindenbaum and A. Tarski (see [Tarski 1935]), and it is clearly an im portant step tow ards a general representation theorem. However, it still leaves us powerless to deal with Boolean algebras which are not atom ic; some new idea is needed. At this point there enters the figure of M arshall Stone. Significantly, Stone was neither an algebraist nor a logician; his m ain w ork had been in functional analysis, with the study of linear operators in H ilbert space [1932]. It was his work in this area, on the spectral resolution theorem, which led him to the consideration of algebras o f commuting projections in H ilbert space; it was know n that these could be given the structure of Boolean algebras, but they had no natural representations as algebras of subsets. The representation theorem thus became a tool of practical im portance to Stone; at the same time, his background in functional analysis gave him a greater familiarity with the new m ethods being developed in general topology than was available to m ost algebraists. (In [1938], Stone sums up his attitude: "A cardinal principle of m odern m athem atical research may be stated as a maxim: “One m ust always topologize” ’. But some algebraists were slow to learn this maxim: in 1946 [H ochschild 1947], G arrett Birkhoff was content to define algebra as ‘dealing only with operations involving a finite num ber of elements’; and when challenged by Artin, M ac Lane and others on the im portance of topological methods, he replied T don’t consider this algebra, but this doesn’t mean that algebraists can’t do it’. Incidentally, Birkhoff had

Introduction

xv

independently arrived at a representation theorem for distributive lattices [1933] which was equivalent to Stone’s; but because he missed the topological significance of the theorem, his version had far less influence on the later development of the subject.) Stone’s w ork on Boolean algebras was published in two long papers in the Transactions of the American M athem atical Society [1936, 1937], though summaries of some of his results had appeared earlier [1934, 1935]. There were two key ideas in Stone’s w ork: one was his realization (described vividly by M ac Lane [1981]) th at a Boolean algebra is the same thing as a particular sort of ring (namely, one in which every element a satisfies a2 = a). Nowadays, when the equivalence of Boolean rings and Boolean algebras is som ething that we set as an exercise to undergraduates in their first course on ring theory, it is hard to understand how this fact rem ained undiscovered for so long. (Actually, Stone’s first work in 1932-3 was based entirely on an informal analogy with ring theory, and it was not until 1935 that he realized the connection could be made form al; this necessitated the rewriting of a large part of his w ork on the subject, which explains the delay in publication of his results.) At any rate, the analogy with rings led Stone to a realization of the im portance of ideals (and particularly prime ideals) in lattice theory; it is the set of prime ideals of a Boolean algebra which provides the carrier set for Stone’s representation. (Notice the contrast with the L indenbaum -T arski representation, in which the carrier set is com posed of elements of the Boolean algebra.) Stone’s second key idea was the introduction of topology. He observed that the set of prime ideals of a Boolean algebra can be made into a topological space in a natural way, in which the open sets correspond to arbitrary ideals of the algebra. (Specifically, to an ideal I we associate the open set of all prime ideals which do not contain I.) In this topology, the clopen sets (those which are both open and closed) correspond to principal ideals, and hence to elements of the algebra; so we can recover (an isom orphic copy of) the original algebra from its space of prime ideals. N ow this was a really bold idea. A lthough the practitioners of abstract general topology (notably the Polish school of Sierpinski [1928], K uratow ski [1933], et al.) had by the early thirties developed considerable expertise in the construction of spaces with particular properties, the m otivation of the subject was still geometrical -- the study of subsets of Euclidean space, and spaces constructed therefrom - and (so far as 1 know) nobody had previously had the idea of applying these techniques to the study of spaces constructed from purely algebraic data such as a Boolean algebra.

xvi

Introduction

However, Stone went ahead and did just that. O f course, given any topological space X, the clopen subsets of X are closed under finite union, intersection and com plem entation, and so form a sub-Booleanalgebra of PX . But Stone showed that the spaces which arise as the prime ideal spaces of their Boolean algebras o f clopen subsets can be character­ ized in purely topological terms, as being com pact, H ausdorff and totally disconnected. (He called such spaces 'Boolean spaces’; subsequent authors have chosen to honour him by christening them 'Stone spaces’.) M oreover, any hom om orphism of Boolean algebras gives rise to a continuous m ap in the opposite direction between their prime ideal spaces; and any continuous m ap of Stone spaces gives rise to a hom o­ m orphism in the opposite direction between their clopen-set algebras. The constructions 'prim e ideal space’ and 'clopen-set algebra’ are thus examples of (contravariant) functors; and together they form one of the earliest nontrivial examples of an equivalence o f categories. All this was proved in detail by Stone, although the categorical language in which we now express it was not introduced until the following decade; but Stone’s Theorem was undoubtedly one of the m ajor influences which prepared the m athem atical w orld for the introduction of categories by Eilenberg and M ac Lane [1942, 1945]. At any rate, the meaning of the equivalence was clear: it m eant that any algebraic fact about Boolean algebras could be translated into a topological fact about Stone spaces, and vice versa. The way was thus immediately open for developing applications of Stone’s Theorem in both algebra and topology. In fact the first applications were in topology and functional analysis. Two of them were already present in Stone’s [1937] paper M isconstruction of the maximal compactification of a completely regular space, and his generalization of the W eierstrass approxim ation theorem. The S toneCech compactification was of course discovered independently (see [Stone 1962]) by Stone and by Cech [1937], but the m ethods of the two were substantially different. C edi’s w ork can be seen as a natural extension of the work of U rysohn [1925a] and Tychonoff [1929] on embedding spaces in products; in fact his construction was a relatively simple develop­ ment of that used by Tychonoff in his proof that completely regular spaces are precisely the subspaces of com pact H ausdorff spaces. In con­ trast, Stone’s construction used algebraic properties of the ring C*(X) of bounded continuous real-valued functions on the space X. This in turn raised the problem of characterizing C*(X) in algebraic term s - a problem which was again solved independently by two people, Stone [1940, 1941] working with real-valued functions, and G elfand [1939, 1941] with com ­

Introduction

xvii

plex ones, thus laying the foundations of the im portant subject of Gelfand duality. (See also [K akutani 1940, 1941], [K rein and K rein 1940, 1943], [K aplansky 1947, 1948a], [M ilgram 1949], [A nderson and Blair 1959], [G illm an and Jerison 1960], [H enriksen, Isbell and Johnson 1961], [M ulvey 1978a, 1979a], etc.) Stone’s [1937] construction of his com pactification also contained a detailed proof of its universal property - once again, as noted by M ac Lane [1970], anticipating an im portant trend in category theory. It can thus be considered as a first step in the ‘algebraization’ of the category of com pact H ausdorff spaces, a process brought to a successful conclusion by M anes’ [1969] proof (which relied heavily on Stone’s com pactification) that this category is indeed algebraic in the technical sense. (See also [E dgar 1973], [Semadeni 1974], and [M anes 1980].) A nother direction of application was initiated by Stone in [1937a]; in considering the topological equivalent of the condition of completeness for Boolean algebras, he introduced the im portant notion of extremal disconnectedness. F urther w ork on extremally disconnected spaces [Stone 1949] confirmed their im portance in functional analysis, and led up to the work of G leason [1958], Rainwater [1959], Iliadis [1963], Banaschewski [1967, 1971] and Dyckhoff [1972, 1976] on projective topological spaces - once again, im porting ideas from algebra (in this case, homological algebra) into categories of topological spaces. M ore recently, Johnstone [1979b, 1980a, 1981] has pointed out the sheaftheoretic and logical ideas underlying this connection. In yet another paper published in 1937 [1937b], Stone generalized his representation theorem to non-Boolean distributive lattices, at the same time introducing the non-H ausdorff cousins of Stone spaces which we now call coherent spaces. A lthough these have received less subsequent attention than Stone spaces, the work of H ochster [1969], Priestley [1970, 1972] and Joyal [1971, 1971a] is w orth mentioning. In another direction, [Stone 1937b] paved the way for the study of topological concepts from a lattice-theoretic viewpoint, initiated by W allm an [1938] and pursued by M cKinsey and Tarski [1944], N obeling [1954], Lesieur [1954], Ehresm ann [1957], Papert [1964], D ow ker and Papert [1966], Isbell [1972] and Simmons [1978], am ong others. (A close relative of this line of development is the study of topological posets and lattices: [F rink 1942], [N achbin 1950], [W ard 1954], [A nderson 1959, 1961, 1962], [Strauss 1968], [Choe 1969], [Law son 1969, 1970, 1973], [Scott 1972], [H ofm ann and Stralka 1976], [Semilattices 1980], etc.) In recent years, the lattice-theoretic approach to topology has merged

xviii

Introduction

in the study of general sheaf theory and topos theory. The origins of sheaf theory [Leray 1945, 1946], [C artan 1949] owe little if anything to the w ork of Stone; but the generalized sheaf theory pioneered by G rothendieck and his followers around 1963 [G iraud 1963], [Verdier 1964], and still m ore the elem entary theory of toposes introduced by Lawvere and Tierney in 1970 [Lawvere 1971], [Tierney 1973], have increasingly focused attention on the fact that the im portant aspect of a space (from a sheaf-theoretic point of view) is not its set of points but its lattice of open subsets. (This is not the place for a detailed history of sheaf theory or topos theory; we refer the reader to [G ray 1979] and the Introduction to [Johnstone 1977].) A nother area where the influence of Stone’s w ork has been strongly felt is the representation theory of rings and m ore general algebraic systems. The foundation-stone of this theory is BirkhofFs subdirect de­ com position theorem [1944], which displays none of the influence of Stone’s topological ideas, but it was soon realized [Jacobson 1945], [Arens and K aplansky 1948] that much sharper representation theorem s could be obtained by introducing topologies in the fashion of Stone’s Theorem. F urther im portant w ork in this direction was done by G illm an [1957], Henriksen and Jerison [1965a], Pierce [1967], D auns and H ofm ann [1968], Keimel [1971], H ofm ann [1972], Davey [1973] and C ornish [1977]; in recent years this line, too, has developed strong links with topos theory [M ulvey 1974,1979], [K ennison 1976], [Tierney 1976], [Johnstone 1977a], [Coste 1979]. It rem ains to consider two areas of m athem atics in which, like the dog in the night-time [D oyle 1892], the influence of Stone’s Theorem is more conspicuous by its absence than by its presence. O ne of these is category theory itself. M ac Lane [1970] has pointed out how the categ­ orical ideas present in Stone’s [1937] paper were not directly followed up by the founders of category theory: in particular, the notion of adjoint functor, though present implicitly in Stone’s description of his com ­ pactification, and strongly suggested by the functional-analytic notion of adjoint operator, was not explicitly introduced into category theory until 1958 [K an 1958]. Stone himself [1970] has analysed the reasons for this failure, pointing out that the algebraic and algebraic-topological back­ ground of the pioneers of category theory naturally m eant th at the protocategorical ideas arising from general topology and functional analysis did not form a part of their experience. (However, it should not be thought that Stone’s w ork has had no influence on category theory. There is one area in particular - the categ­

Introduction

xix

orical study of duality theorem s - which, whilst it also owes a good deal to the duality theory of Pontryagin [1934] and van K am pen [1935], draws a very large part of its inspiration from the duality theorems of Stone and Gelfand already mentioned. F o r w ork in this area, see [H ofm ann 1970], [Isbell 1972a, 1974], [H ofm ann, Mislove and Stralka 1974], [K eim el and W erner 1974], [L am bek and R attray 1978, 1979], [Law son 1979] and [B arr 1979].) The other area where one searches in vain for the influence of Stone’s Theorem is in algebraic geometry, with the rise of the ‘Zariski topology’. It was sometime in the late forties (see [Zariski 1952]) th at O. Zariski realized how one m ight define a topology on any abstract algebraic variety, by taking its algebraic subsets as closed sets; the precise date is difficult to determine, since Zariski himself does not seem to have attached m uch im portance to the idea. (There is no m ention of the Zariski topology in the first edition of Weil’s book [1946] on algebraic geometry, although it plays a central role in the second edition [1962].) It was not until the work of Serre [1955] that the Zariski topology became an im portant tool in the application of topological m ethods (in this case, sheaf cohomology) to abstract algebraic geometry. There is an obvious similarity between the topologies introduced by Zariski and Stone, and indeed Dieudonne [1974] asserts that Zariski was influenced by Stone’s w ork; but there seems to be no acknowledgem ent of this influence in Zariski’s own papers. The refoundation of algebraic geometry using schemes in place of varieties, begun by G rothendieck [1959, 1960] in the late fifties, brought the Zariski and Stone topologies even closer together: indeed, the latter is ju st the special case of the former applied to the spectrum of a Boolean ring. But again, one will not find any reference to Stone in the w ork of G rothendieck, even though his use of the w ord ‘spectrum ’ is an obvious echo of [Stone 1940], and G rothendieck, with his background in func­ tional analysis, m ust have been familiar with Stone’s work in that field. Again, when the Zariski topology made its first appearance in a book on com m utative algebra, as opposed to algebraic geometry, [Bourbaki 1961a], there was no m ention of Stone’s name. (The Zariski topology does not occur in [Zariski and Samuel 1958].) O ne area which has not been m entioned in this survey is m athem atical logic. This is not because Stone’s work has failed to have an influence here, but because until recently (if one discounts such papers as [£ o s and Ryll-Nardzewski 1954]) the full extent of th at influence has rarely been m ade explicit. It is only since the rise of elementary topos theory, and the consequent interest in coherent logic ([Reyes 1974], [M akkai and Reyes

xx

Introduction

1977]) and sheaf models ([Scott 1968, 1970], [M acintyre 1973], [C om er 1974], [Loullis 1979], [Burris and W erner 1979]), th at m athem atical logic’s debt to Stone has become apparent. (In a different direction, however, the Stone-C ech com pactification has inspired a good deal of set-theoretical work on ultrafilters - see [W alker 1974], [C om fort and N egrepontis 1974], [C om fort 1977, 1980], etc.) W hat conclusions should we draw from this historical survey? First, it emphasizes the absurdity of attem pting to divide m athem atics into w atertight com partm ents. We have seen th at Stone’s w ork has had an influence on alm ost every area of m odern m athem atics, with the exception of finite group theory and com binatorics on the one hand, and of classical analysis and num ber theory on the other. M oreover, although in the survey I have necessarily tried to classify the papers to which I have referred into a num ber of separate lines of development, even a brief perusal of a few o f them will show the extent to which they defy such classification. In particular, it should be clear th a t any attem pt, as in [K uyk 1977], to draw a distinction between 'discrete' and 'continuous’ m athem atics belittles the subject: abstract algebra cannot develop to its fullest extent w ithout the infusion of topological ideas, and conversely if we do not recognize the algebraic aspects of the fundam ental structures of analysis our view of them will be one-sided. (For further com m ents on the relationship between discrete and continuous mathem atics, see [Lawvere 1975].) A second, related, point concerns the danger of adopting a narrowly specialist approach to mathematics. The enorm ous increase in the num ber of practising m athem aticians since the 1930s has inevitably produced a corresponding decrease in the range of m athem atical knowledge that each one possesses on average, and the effect of this is easy to see: theorems and techniques which are com m onplace in one field are laboriously and imperfectly rediscovered in adjacent ones. (The Bibliography of this book contains more examples of this phenom enon th an I w ould care to count.) In contrast, Stone stands as an example of a m an who, although his interests may lie in one particular area of mathem atics, has nonetheless a sufficiently general perspective on the whole subject to recognize the significance o f his w ork for other fields; some of his successors have un­ fortunately been less broad in their outlook. I therefore hope that this book, by dem onstrating the fundam ental unity of ideas from many differ­ ent areas of mathematics, may help at least some of its readers in develop­ ing this breadth of vision. (I should add that, although the writing of the book has undoubtedly assisted its author in this way, he is still acutely

Introduction

xxi

aware of his own shortcom ings; I crave the indulgence of all specialists in such fields as functional analysis, ring theory and topological algebra for the num ber of times my ignorance of their subjects shows through.) Finally, I believe the legacy of Stone’s w ork shows us that it is w rong to regard the ‘soft’ or conceptual side of m athem atics as inferior to the 'h ard’ or com putational side. Stone’s own w ork is all in the direction of abstraction and conceptualization, but it has produced substantial benefits in many areas of m athem atics - benefits which could only have been gained from the linking o f ideas in an abstract framework. (See, in this context, the article of N ewm an [1942]; also [D ieudonne 1975] and [M ac Lane 1980] for two contrasting views of the way in which m athem atical p ro ­ gress arises from conceptual development.) U nfortunately, lim itations of space mean that this book is concerned alm ost entirely with theory and very little with applications - especially in the later chapters, where we shall frequently have to leave a particular topic ju st at the point where the h ard com putational work begins - thereby providing free am m unition for those (such as Linderholm [1971]) who delight in sneering at 'abstract nonsense’. I therefore urge my readers not to regard this book as an end in itself, but rather to go on and use it as a foundation on which to build whatever m athem atical edifice may suit their own tastes and interests.

I

Preliminaries

1. Lattices 1.1 Let A be a set. A partial order on A is a binary re la tio n ^ which is (i) reflexive: for all a e A, a ^ a , (ii) transitive: if a ^ b and b ^ c , then and (iii) antisymmetric: if a ^ b and b ^ a , then a — b. A poset (short for partially ordered set) is a set equipped with a partial order. 1.2 Let A be a poset, S a subset of A. We say an element a e A is a join (or least upper bound) for S, and write a — VS, if (i) a is an upper bound for S, i.e. s ^ a for all s e S, and (ii) if b satisfies Vs e S (s^ b \ then a ^ b . The antisymmetry axiom 1.1 (iii) ensures that the jo in of S, if it exists, is unique. If S is a two-element set {s, t}, we write s v t for V {s, f ] : and if S is the empty set 0 , we write 0 for V 0 - clearly 0 is just the least element of A. 1.3 Let A be a poset in which every finite subset has a join. Then the binary operation v and the element 0 defined above satisfy the equations (i) a v a = a (ii) a v b = b v a (iii) a v {b v c) = (a v b) v c (iv) a v 0 = a for all a, b, c. Briefly, we can say that (A , v , 0) is a com m utative m onoid (semigroup with unit) in which every element is idempotent. Conversely, we have 1

2

I: Preliminaries

Theorem Let (A , v , 0) be a com m utative m onoid in which every element is idempotent. Then there exists a unique partial order on A such th at a v b is the join of a and b, and 0 is the least element. Proof Clearly, if such a partial order exists, we must have a ^ b iff a v b — b. Conversely, taking this as a definition of we deduce reflex­ iveness of ^ immediately from equation (i), and antisymmetry from the form of the definition. To show transitivity, suppose a ^ b and b ^ c . Then a v c ~ a v ( b v c ) since b ^ c —( a v b ) v c

by equation (iii)

=bvc

since a ^ b

—c

since b ^ c ,

so a ^ c . N ow let a, b be any two elements of A. Then a v{a v b ) =( a v a ) v b = a v b, so a ^ a vft, and similarly (using equation (ii)) b ^ a v b. But if a ^ c and b ^ c , then (a v b ) v c —a v (b v c ) ~ a v c — c, so a v b ^ c ; i.e. a v b is the join of a and b. Finally, equation (iv) says immediately th at 0 is the least element of A. □ A set with the structure described in the Theorem is called a semilattice (sometimes joi&semilattice). The Theorem says th a t the notion of semi­ lattice can be defined either in terms of the order relation or in term s of the jo in operation; but when we come to consider hom om orphism s (structurepreserving maps), there is an im portant difference. A semilattice hom o­ m orphism / \A-+B (i.e. a m ap preserving the distinguished element 0 and the operation v ) is necessarily an order-preserving map, but an order-preserving m ap between semilattices need not be a hom om orphism . Exercise Give an example of a semilattice A and a subset B g A which is a semilattice in the induced ordering, but not a sub-semilattice of A.

1.4 Dually, in any poset we can consider the notion of meet (greatest lower bound), defined by reversing all the inequalities in the definition of join. We write AS, a / \ b and 1 for the analogues of VS, a v b and 0. A lattice is a poset A in which every finite subset has both a join and a meet; by Theorem 1.3, this is equivalent to saying th at A is equipped with two binary operations v , a and two distinguished el­

3

Lattices

ements 0, 1 such that both (A , v , 0) and (A, a , 1) are semilattices, and the partial orders on A induced by the two semilattice structures are opposite to each other. Proposition Suppose (A , v , 0) and (A , a , 1) are semilattices. Then (A , v , a , 0, 1) is a lattice iff the absorptive laws a a {a v b) = a,

av{aAb) = a

are satisfied for all a, b e A. Proof Suppose the absorptive laws are satisfied. Then a v b = b implies a a b = a a (a v b) = a, and conversely. So the two partial orders on A agree. The converse is easy. □ O u r formal definition of a lattice is thus ‘a set with two binary operations v , a and two distinguished elements 0,1, such th at v (respectively a ) is associative, com m utative and idem potent and has 0 (respectively 1) as unit element, and such that v and a satisfy the absorptive laws’. Remark M any authors define a lattice purely in terms of v and a , w ith­ out requiring the existence of 0 and 1. This seems unnatural, since if a poset has joins and meets for all nonem pty finite sets it is reasonable to require them for the empty set as well; and m ost of the examples we meet will bear this out. O n the other hand, we don’t require the elements 0 and 1 to be distinct; we do wish to consider a one-elem ent set, with its unique partial ordering, to be a lattice. 1.5 in most of the lattices we’ll encounter, the operations v and a will satisfy an additional identity, namely the distributive law (i) a a (b v c) = (a a b) v (a a c ) for all a, b, c. (If we think of a as m ultiplication and v as addition, this is just the distributive law of ordinary arithmetic.) Lemma If the distributive law holds in a lattice, then so does its dual, i.e. the identity (ii) a v (b a c) = {a v b) a (a v c).

4

I: Preliminaries Proof We have (a v b) a (a v c) = ((a v b ) = a v({a = (a

v

a

a) v ((a v b )

a c

)

v(b

a

c)

ac))

(a a c)) v (b a c)

= av{b

a

c)

applying (i) twice and the two absorptive laws.

□

N ote also that in the presence of (i) we can deduce either absorptive law from the other, since we have a a (a v b) = {a a a) v (a a b) = a v (a a b).

1.6 Proposition Let a, b, c be three elements of a distributive lattice A. Then there exists at m ost one x e A satisfying x a a = b and x v a = c. Proof Suppose x and y both satisfy the conditions. Then x = X A (x v a ) = x A c = xA (_yva) = (x A y ) v ( x A a )

by distributivity

= (xAy)vb = XAy since b = x A a = y y = x Ay; so x = y.

Aa

is a lower bound for {x, y}. Similarly we have

□

In any lattice, an element x satisfying x a a = 0 and x v a — 1 is called a complement of a. The Proposition tells us that in a distributive lattice complements are unique when they exist. A Boolean algebra is a distri­ butive lattice A equipped with an additional unary operation “ I :A-+A such that "1 a is a complement of a. Since “ 1 is uniquely determ ined by the other data in the definition, it follows th at any lattice hom om orphism f \ A - * B between Boolean algebras is actually a Boolean algebra hom o­ m orphism (i.e. commutes with ~~1). 1.7 It’s time we had some examples. {a) For any set X , the set P X of all subsets of X is a lattice, with ^ interpreted as inclusion, v and a as union and intersection of subsets, and 0 and 1 as the empty set and the whole of X. M oreover, P X is distributive, since for subsets A , B, C o f X we have

Lattices

5 x e ^ n ( 6 u C ) o x e A, and either x e B or x e C either x e A and x e B, or x e A and xeC « x e ( AnB)Kj (AnC),

so that ^ n ( B u C ) = (i4 riB )u (A n C ). And P X has com ­ plements for all its elements, so it is a Boolean algebra. (,b) Let A be a totally ordered set with least and greatest elements 0 and 1. Then A is a lattice, with v and a interpreted as max and min. We leave it to the reader to verify th at A is distribu­ tive. But if A has m ore than two elements, it is not a Boolean algebra; for no element other than 0 and 1 can have a com p­ lement. (c) Let G be a group. The set of subgroups of G, ordered by inclusion, is a lattice in which meet is again interpreted as intersection, but the join of two subgroups is the subgroup generated by their union. This lattice is not in general distribu­ tive; for example if G is the non-cyclic group of order 4, the lattice looks like

(where a, b, c are the three subgroups of order 2), and each of a7 b, c has two distinct complements. 1.8 Next, we sketch the equivalence between Boolean algebras and Boolean rings, in any Boolean algebra A, we define the symmetric difference operation + by a + b = (a a ~ \b )v(b

a

"1 a).

Lemma The distributive law a A(b + c) = (a Ab) + (a a c ) holds. Proof First note that since “ 1 is an order-reversing bijection of A onto itself, it carries joins into meets and vice versa; that is, the De Morgan laws ~\(a A b ) = ~ \ a v ~\b hold. N ow we have

and ~\ ( a v b ) = ~\aA ~\b

6

I : Preliminaries (a A b ) + (a

Ac)

= {a A b a "1(a a c)) v ( a

= ( f l A f ? A ( 1 a v "1c)) v ( a = ((a A b

a

= (0 v ( a

Ab a

=

(a a

=aa

~\a) v ( a

b a ~\c) ((b a

Ab a

a c

c a

~\(a

Ab))

a ( “ la v "1b ) )

~~lc)) v ((a a c a “ la) v (a a c a

~~lc)) v (0 v { a a c a v (a a

a c a

~\b))

~\b))

~~1b )

~I c) v (c a ~~I b) )

— a A(b + c).

□

W e leave the verification o f the associative law a + (b + c ) = ( a + b) + c

as an exercise for the reader. N ow for any a, we have a + a = (a a ~~Ia) v (a

a

Ia) = 0 v 0 = 0

and

fl + 0 = ( f l A l ) v ( 0 A ~ \a ) ~ a v 0 = a. So (A , + , 0) is a group (clearly commutative, by the definition o f + ), and (A , + , a , 0, 1) is a com m utative ring with 1. 1.9

Conversely, let A be a ring with 1 in which every element

satisfies a2 = a. (We call A a Boolean ring.) Then Lemma (i) A is commutative. (ii) Every a e A satisfies a + a = 0. P roo f Consider a + b = {a + b)2 = a 2 4- ab + ba + b2 = a + b + ab + ba. So ab + ba = 0. Putting a = b, we get a + a = 0; hence a b = —ba and we have ab = ba.

□

So the multiplicative structure (A, *, 1) is a semilattice, with partial order defined by a ^ b iff a b ~ a (cf. Theorem 1.3). It is clear that 0 is the least element of A for this order. N ow consider a + b + ab. We have a(a + b + ab) = a-\-ab-\-ab = a and b(a + b + ab) = ba + b + ab = b,

1

Lattices

so a + b + ab is an upper bound for {a , b}. But if c is an upper bound for { a , b}, then (a + b + ab)c —ac + bc + abc = a + b + ab,

so a + b + ab is the least upper bound. Denoting a + b + ab by a v b , we thus have a lattice structure (A, v , *, 0, 1). M oreover, by an argum ent like that of Lemma 1.8, we may verify that * is distributive over v ; and it is also easy to verify th a t 1 f a is a com plem ent for a. So A is a Boolean algebra. W hat is the symmetric difference operation in this Boolean algebra? We have {a a ~~1b ) v { b a ~\a)(a( 1 f b)) v{b( 1 f a ) )

= ( a f ab) v{b-\-ab) = a f ab f b f ab f (a f ab)(b f ab) —a f b f 6 ab = a f b. Thus if we start from a Boolean ring and turn it into a Boolean algebra by the definitions of 1.9, then back into a Boolean ring by 1.8, we recover the original ring. Similarly if we start from a Boolean algebra and go round the other way. M oreover, it is clear from the nature of the con­ structions that any Boolean algebra hom om orphism is also a Boolean ring hom om orphism , and conversely; so we have proved Theorem The category of Boolean algebras is isom orphic to the category of Boolean rings. □ 1.10 A nother class of lattices which we shall encounter is the class of Hey ting algebras. To introduce this concept, let a and b be el­ ements of a Boolean algebra, and consider the element “ la vb. We have c ^ ~ la v b

c

a

a^ a

a

= (a

a

(H a vb) ~la) v (a

a

b)

= 0v(aAb) = aAb ^b;

and conversely c a a^b

~ \ a v b ^ ~ \ a v ( c Aa)

= ( la v c ) A ( l a v a ) = (

Ia vc)

a

1

8

I: Preliminaries

Thus "1 a v b is the unique largest element c satisfying c a a ^ b. A lattice A is said to be a Heyting algebra if, for each pair of elements (a, b), there exists an element {a-+b) such that c ^ (a -» b ) iff c a a ^ b . Lemma Let A be a lattice, -> a binary operation on A. Then -► makes A into a Heyting algebra iff the equations (i) a->a = 1 (ii) a a (a-»b) = a a b (iii) b A ( a - + b ) = b (iv) a - > { b a c ) = (a->I?) a (a->c) hold for all a, b, c in A. Proof Suppose the equations hold. Then if c ^ (a -* b ), we have c a a ^ a a (a—>b) ~ a a b ^ b. Conversely, if c A a ^ b , then c —c a(a-> c) ^(a->a)

a

by (iii)

(a->c)

~a^{aAc)

by (i) by (iv)

^ a -> b since (iv) implies that the m ap a-> (-) is order-preserving. N ow suppose A is a Heyting algebra. Since we always have a A c ^ a , it is clear that a~>a=l. And since b A a ^ b , we have b ^ a -> b , i.e. b A(a^>b) = b. Now a A{a^>b)^b by definition, and aA (a->fo)^a, so a A ( a - » b ) ^ a A b . But ( a A b ) A a ^ b , so a A b ^ a ^ b , and a A b ^ a , so a A b ^ a A(a~>b). Finally, it is clear that a -> (-) is order-preserving, so a->(b A c)^(a->b)A (a->c). But (a->b) a (a-+c) A a ~ ( a A(a->b)) a (a a (a->c)) ^ b AC,

so (a->b)

a

(a->c)^a->(b

a c ).

□

1.11 in a Boolean algebra A, we can recover the unary operation “ 1 from the binary operation since ~~la = (a->0). In a general Heyting algebra, we take this as the definition of ~~1, and call “ la the negation (or pseudocomplement) of a. It is clear that a a ~\a = 0 (in fact “ la is the largest element of A with this property), but in general we do not have a v ~\a = 1. Lemma (i) A Heyting algebra is distributive.

Lattices

9

(ii) A Heyting algebra A is a Boolean algebra iff ~\~\a = a for all a eA. Proof (i) Since a a ( - ) is order-preserving, we have (aAb) v {a a c) ^ a a (b v c). But a-+((a a b ) v ( a A c))^(a-> (a Ab)) v(a-*-{aAc)) ^ b vc, so that (a a b) v (a a c) ^ a a (b v c). (ii) If is a Boolean algebra, then the identity "1 ~\a —a is clear from uniqueness of complements (Proposition 1.6). Conversely, suppose “ 1“ 1a —a holds in a Heyting algebra A ; since we know A is distributive, we need only verify the identity a v ~ \ a = 1. But the given condition implies that 1 is a bijection A-+A, and it is clearly order-reversing, so the De M organ laws hold (cf. Lemma 1.8). Thus on negating the equation a a "~la = 0, we obtain ~ \ a v ~ \ ~ \ a — ~ \ a v a = \ . □ Exercises (i) Show that the law ~\{a v b)= ~~1a a ~~1b holds in any Heyting algebra, though its dual may fail. (ii) Show that “ 1~~l(a a b)= ~~l ~\a a ~~1 ~~1b. (iii) Show that the ternary operations and p2 defined by p t (a, b, c) = ((a->b)->c) A((c->b)->a) p 2(a, b, c) = (b-»(a a c ) ) a ( a vc) satisfy the equations p(a, a,b) = b and p(a, b, b) = a. (An oper­ ation satisfying these equations is called a Mal'cev operation; see [Sm ith 1976].) Does p t (a, b, c) = p2(a, b, c) in general? (iv) (only for the foolhardy) Give a purely equational proof (i.e. one which doesn’t m ention the order relation) of the dis­ tributive law from the axioms of Heyting algebras (i.e. the axioms of lattices plus the four axioms of Lem ma 1.10). 1.12 We have seen that the class of Heyting algebras includes the class of Boolean algebras, and is included in the class of distributive lattices. We need a couple of examples to show th at both inclusions are strict. (a) Let A be a totally ordered set as in Example 1.7(b). Then A is a Heyting algebra, with im plication defined by a ^ b = 1 if a ^ b =b

otherwise.

10

I : Preliminaries But A is not normally B oolean; in fact every a =£ 0 in A satisfies (b) Let X be an infinite set, and let A be the subset of the powerset P X consisting of all finite subsets of X together with X itself, it is easy to see that A is a sublattice of P X , and therefore distributive. But A is not a Heyting algebra, for if a is a n o n ­ empty finite subset of X , the set of members of A having empty intersection with a has no largest member. Exercises (i) Show that if we reverse the ordering in the lattice of Example (b\ we do get a Heyting algebra. Thus the concept of Heyting algebra, unlike those of distributive lattice and of Boolean algebra, is not self-dual. (ii) Show that every finite distributive lattice is a Heyting algebra. [Consider the join of all elements c satisfying a Ac^b.~\

1.13 Given a Heyting algebra A , we say a e A is regular if “ I “ la —a. The set of all regular elements of A , with its induced order, is denoted A-\-\. Proposition A-\-\ is a Boolean algebra (though it is not in general a sublattice of A). Proof Since “1 0= 1 and “ 11 = 0 , 0 and 1 are in A ^ . And since “ 1“ 1 com m utes with a , A-\-\ is a sub-meet-semilattice of A. To define joins in A note first that a A “ la = 0 implies —la. Hence ~~la^ “ 1“ 1“ 1a; but since “ 1 is order-reversing we also have ~~la^ “ i n i a , i.e. “ la e A -\-\. Now it is clear that “ 1“ l(a v Ab)is a least upper bound for a and b in A-\-\. Distributivity of A ^ again follows from the fact th at “ I “ 1 commutes with a . The com plem entation operation in A n -\ is simply “ 1^; it remains to show that a v A^ 1 a = 1, i.e. “ 1“ l(a v ^ “ la) = 1. But by Exercise 1.1 l(i) we have 11av1aH 1(1aA l1a)= 10= l.

□

Exercise Show that the following conditions on a Heyting algebra A are equivalent: (i) De M organ’s law “ l(a a b)= “ 1a v “ 1b holds. (ii) The identity “ la v “ I “ la = 1 holds.

11

Ideals and filters (iii) Every element of has a com plem ent in A. (iv) >4-i-i is a sublattice of A. 2. Ideals and filters

2.1 The identification of Boolean algebras with certain rings, established in paragraph 1.9, allows us to im port ring-theoretic ideas into Boolean algebra, and into lattice theory in general. In this section we investigate the lattice-theoretic analogues of ideals and prime ideals. A subset / of a join-sem ilattice A is said to be an ideal if (i) / is a sub-join-semilattice of A; i.e. 0 e l , and a e l , b e l imply a v b e l ; and (ii) / is a lower set; i.e. a e l and b ^ a imply b e I. Example F or any a e A , the subset |( a ) = {b e A \ b ^ a } is an ideal of A; and it is clearly the smallest ideal containing a. So by analogy with ring theory, we call it the principal ideal generated by a. (We shall also use the notation |(a ) for {b e A \b ^a } .) Lemma (i) Let / :A^>B be a semilattice hom om orphism . Then the set {a e A\ f {a) —0} (the kernel of / ) is an ideal of A. (ii) Let / be an ideal of a join-sem ilattice A. Then there exists a semilattice hom om orphism / :A-+B with kernel /. (iii) If, in (ii), A is a distributive lattice, then we may take B to be a distributive lattice and / to be a lattice hom om orphism . Proof (i) is trivial. (ii) Define an equivalence relation = , on A by a = j b iff there exist j e l such that a v i — b v j . It is easily verified th at this is an equivalence relation, and that a = j b implies a v c = f b v c for any c, so that we can make the ~ r classes into a semilattice B, in such a way th at the canonical projection A^>B is a hom om orphism . Now the kernel of this projection is the = r class of 0, which is clearly /. (iii) In this case we have to verify that a = j b implies a A c = j b a c , s o that B can be made into a lattice (clearly distributive, since it is a quotient of A). But if a v i = b v j , then (a v i) and i a c ,

j ac

ac

—(a a c ) v

(/ a

c) = {b a c) v

e I since / is a lower set.

{ j a c) ,

□

12

/• Preliminaries

Exercise Show that the assum ption of distributivity is needed in (iii), by exhibiting an ideal in the lattice of 1.7(c) which is not the kernel of any lattice homom orphism . 2.2 in contrast to the ring-theoretic case, a surjective (semi)lattice hom om orphism is not determ ined by its kernel. F o r example if A = {0, a, 1} is a three-element totally ordered set, there is a lattice surjection from A to the two-element lattice {0, 1} having kernel {0}, the same as the identity m ap on A. To determine the surjective part of a lattice hom om orphism / :A^>B, we thus have to look at the inverse images of other elements of B besides 0. in particular, we may consider { a e A \ f ( a ) = i } , which clearly satisfies axioms dual to those defining an ideal; we call such a subset of A a filter. Proposition Let / be an ideal of a lattice A. The following conditions are equivalent: (i) The complement of / in A is a filter. (ii) 1 £ /, and (a a b e I) implies either a e l or b e l . (iii) / is the kernel of a lattice hom om orphism / :A->2, where 2 denotes the two-element lattice {0, 1}. Proof (i)=>(ii): Let F denote the com plem ent of I. Since F is a filter, we have 1 e F and so 1 £ /. Similarly, a a b e I implies not (a e F and b e F \ whence a e I or b e I. (ii)=>(iii): Define / by f (a) = 0

if a e /, f ( a) = 1

otherwise.

We have to verify that / is a lattice hom om orphism ; but this is easy. (iii)=>(i) since the complement of / is the filter f ' l ( 1).

□

An ideal satisfying the equivalent conditions of the Proposition is called a prime ideal; its complement (which of course satisfies conditions dual to those of (ii)) is called a prime filter. Exercise Verify that a subset of a Boolean algebra A is an ideal (prime ideal) of A considered as a lattice iff it is an ideal (prime ideal) of A con­ sidered as a ring.

13

Ideals and filters

2.3 We now em bark on the fundam ental existence theorem for prime ideals, it should be noted that this theorem makes essential use of the axiom of choice (in the form of Z orn’s Lemma). We shall adopt the habit of m arking with an asterisk all those lemmas, propositions, etc., whose proofs depend on some form of the axiom of choice. (The extent of this dependence will be discussed in the Notes.) *Lemma Let / be an ideal of a lattice A , and F a filter disjoint from /. Then there exists an ideal M of A which is maximal am ongst those containing / and disjoint from F. Proof Apply Z orn’s Lemma to the set of ideals which contain / and are disjoint from F. To justify this application, we have only to verify th at the union of a family of ideals which is totally ordered by inclusion is again an ideal; but this is straightforw ard. □ 2.4 T heorem Let F be a filter in a distributive lattice A , and / an ideal which is m axim al am ongst those disjoint from F. Then / is prime. Proof Since / n F = j 0 ', 1 £ / . Suppose we have a i A a 2 * I \ con­ sider the ideals J u J 2 generated by / together with a u a2 respectively, it is not hard to see that a typical mem ber of J, has the form i v (at a b) with i e I , b e A (for the set of all elements of this form is an ideal). Suppose that J j and J 2 both meet F ; then we can find i u i2i b u b2 such that ix v (ax a b t ) and i2 v ( a 2 a b2) are both in F. But F is a filter, so it also contains (ix v ( a l A b 1))A(i 2 v ( a 2 A b 2)) = (ii a

i 2)

v(/x

A a 2 A b 2) v ( i 2 A a i A b i ) v { a i a q 2 A b i a b 2\

which is in / since each of the four terms in the disjunction is in /. But this contradicts I n F —0 ; so at least one J t is disjoint from F. But J f 3 / , so by maximality we have J,- = /, and hence ax e / . So we have verified condition (ii) of Proposition 2.2. □ We norm ally apply the Theorem in the case when F is the m inimal filter {1}, so that / is a maximal proper ideal. The term 'maximal ideal’, w ithout further qualification, will invariably mean 'm axim al proper ideal’, so that we can state

14

I: Preliminaries Corollary in a distributive lattice, every maximal ideal is prime.

□

Exercise Show that the distributivity hypothesis cannot be om itted from the Corollary. [Use Example 1.7(c) again.] 2.5 Com bining the results of the last three paragraphs, we obtain a proposition which contains in essence all th at we need to prove the Stone Representation Theorem. * Proposition Let a and b be elements of a distributive lattice A , with a ^ b . Then there exists a lattice hom om orphism / : A ->2 with f ( a) = 0 and f(b)=l. Proof Take /, F to be the principal ideal J,(a) and the principal filter j (b\ respectively. The condition a ^ b ensures that / and F are disjoint, so by Lemma 2.3 we can enlarge / to a maximal / ' disjoint from F. By Theorem 2.4 /' is prime, so by Proposition 2.2 it is the kernel of a hom om orphism / M - > 2. But then a e l ^ l \ so f (a) = 0; and b e F , so b t r and f ( b ) = l . □ * Exercise Use Proposition 2.5 to show that any distributive lattice A is isomorphic to a sublattice of P X for some set X . [Take X to be the set of hom om orphism s A->2.] 2.6 F o r Boolean algebras, we have a converse to Corollary 2.4: Proposition Let / be an ideal of a Boolean algebra A. The following conditions are equivalent: (i) / is prime. (ii) F or every a e A, just one of the pair {a, “ la} is in /. (iii) / is maximal. Proof (i)=>(ii): Since a a H a = 0 e /, we m ust have at least one of {a, "1 a} in I. But since a v "1 a = 1 £ /, we cannot have both. (ii)=>(iii): Since 0 e /, we have 1 = “ 10 £ /, so / is proper. Let J be an

Some categorical concepts

15

ideal strictly containing /, a e j —I. Then "1 a e l , so l = a v " l a e J , i.e. J is improper. So / is a maximal proper ideal. (iii)=>(i) is Corollary 2.4. □ The assum ption of Booleanness in the Proposition is essential. For example, if A is a totally ordered set, then every proper ideal of A is prime, but only one is maximal. Indeed, as a consequence of the Stone Repre­ sentation Theorem, we shall obtain a converse to Proposition 2.6: if every prime ideal of a distributive lattice A is maximal, then A is a Boolean algebra (Corollary il 4.9). 3. Some categorical concepts 3.1 This section is not intended to provide a comprehensive introduction to category theory for the beginner; such an introduction would require more space than we can possibly afford to give it. O ur aim is rather to delineate the territory we shall regard as familiar, by giving the statem ents of the m ajor results we shall be assuming later on, and providing references (mostly to the [1971] book of M ac Lane) for their proofs. We begin with the definitions of the three fundam ental concepts. (a) A category C consists of three things: (i) A class of objects, usually denoted by capital letters A, B, C , . . . (ii) A class of morphisms, usually denoted by lower-case letters f g , h , . . . Each m orphism has a domain and codomain which are objects of C; we write for J is a m orphism and dom ( f ) — A and cod (/ ) = B \ (iii) A composition law which assigns to each pair of m orphism s (f g) with d o m (/) = cod(A satisfying and idA g = g whenever the composites are defined. (b) A functor T : C -»D is a m orphism of categories. Specifically, it consists of functions (objects of C)-»(objects of D) and (morphisms of C)->(morphisms of D), both denoted T, such that

16

I: Preliminaries (1) if f then T ( f ) : T(A)-> T(B). (ii) T ( f g ) = T { f ) • T(g) whenever fg is defined. (iii) T{\dA) = \dT{A) for all A. (c) A natural transformation a : S - > T between two functors S, T : C->D consists of a function (objects of C)-»(morphisms of D), denoted A such that (i) ctA :S(A)-+T(A) for a lM . (ii) For all / : A ^ B in C, we have T ( f ) *ola = olb • S( f ) : S(A) -T (B ).

3.2 We shall be concerned mainly with concrete categories whose objects are sets with some kind of structure and whose m orphism s are structure-preserving functions, the com position law being the usual com ­ position of functions. O ur usual convention will be to name such a categ­ ory by (an abbreviation of) the com m on name of its objects; thus we have Set = category of sets and functions. Sp = category of topological spaces and continuous maps. Pos = category of partially ordered sets and order-preserving maps. L at = category of lattices and lattice hom omorphisms. S L at = category of semilattices and hom om orphism s. D L at = category of distributive lattices and hom om orphisms. Bool = category of B oolean algebras and hom om orphism s. CRng = category of com m utative rings (with 1) and hom o­ morphisms. F urther examples will occur frequently as we go along. We shall occasionally have to consider pairs of categories having the same objects but different m orphism s; in this case we shall adopt the slightly unusual custom of giving two different names to the objects, which are entirely synonym ous when we refer only to objects, but m ean different things when applied to morphisms. i hope that this procedure will actually be less confusing than it sounds. We note that a concrete category C is always locally small; that is, for each pair of objects (A, B), the m orphism s A-> B in C form a set hom c (A , B) (or simply hom (A, B)), rather than a proper class. 3.3 The other class of categories which we shall frequently meet is th at consisting of categories arising from partially ordered sets. If (A , ^ ) is a poset, we can make it into a category A whose objects are the elements of A and whose m orphism s are the instances of the order-relation - i.e.

Some categorical concepts

17

there is just one m orphism a—>b if a ^fr, and none if a =?£b. It is clear that transitivity of ^ ensures the existence of a unique com position law for A, and reflexivity ensures the existence of identities; thus we can consider posets as a special case of categories. It turns out th at many of the basic concepts of category theory become, when we specialize them to posets, concepts already familiar in lattice theory - for example, a functor between posets is just an order-preserving map.

3.4 The central concept of category theory is the notion of adjunc­ tion. Given functors F : C->D and G : D->C, we say F is left adjoint to G (and write F -IG ) if there is a bijection, natural in the variables A and B, between m orphism s / :A^>GB in C and m orphism s / : F A ^ B in D. Given an adjunction ( F —IG), we may consider for each A the m orphism tjA : A^>GFA which corresponds to idf ^ : F A ^ F A . N aturality of /(-►/ tells us that rj is a natural transform ation idc-> GF (called the unit of the adjunction); also, since for any / : A^>GB we have / = / * idf ^, we obtain f ~ G ( f ) ' t \ A - i.e. the bijection J v + f is recoverable from knowledge of the functor G and the natural transform ation r\. In fact we don’t even need to know that F is a functor: Proposition (i) A functor G : D ->C has a left adjoint provided we can find, for each object A of C, an object FA of D and a m orphism t\A . A ^ G F A which is ‘universal am ong m orphism s from A to the image of G’ in the sense that for each / : A ^ G B there is a unique / : F A ^ B satisfying f = G ( f ) • rjA. Proof See [M ac Lane 1971], p. 81, Theorem 2(ii).

□

Example Let Gp be the category of groups and group hom om orphism s, and G Gp -►Set the forgetful functor which sends a group to its under­ lying set and a hom om orphism to its underlying function. If X is a set, the defining property of the free group F X generated by X says precisely that the inclusion X ^ F X is universal am ong functions from X to the image of G. So G has a left adjoint, the free functor Set->Gp. F or further examples of adjunctions, see [M ac Lane 1971], p. 85. Yet another description of adjunctions involves both rj and its dual, the counit e : FG->idD(eB= (idGB) : FGB-»B). Since we have / = G ( f ) *y\a and

18

I: Preliminaries

f = e B - F ( /), it is easy to see that rj and e satisfy the com m utative diagram s (‘triangular identities’)

Proposition (ii) Let F : C -»D and G : D -»C be functors, and t] : idc -»G F and s : FG -»idD natural transform ations satisfying the triangular identities. Then there is an adjunction ( F —IG) having unit t] and counit e. Proof See [M ac Lane 1971], p. 81, Theorem 2(v).

□

Corollary Let f : A - + B and g :B^>A be order-preserving maps between posets, and regard them as functors as in 3.3. Then f - \ g iff the inequalities a ^ g f { a ) and f g{ b) ^ b hold for all a e A and b e B. M oreover, if these hold, then f g f = / and g f g ^ g , and / and g restrict to a bijection between the subsets {a\a = gf{a)) and {b\b=fg{b)\ of A and B. Proof Since every diagram in a poset commutes, the given inequalities define natural transform ations y\ and r. as required, and the triangular identities are autom atic. The second part follows from anti­ symmetry, since we have f ( a ) ^ f g f { a ) and f g f ( a ) ^ f { a \ etc. (See also [M ac Lane 1971], p. 93, Theorem 1.) □ A reflection is an adjunction for which the counit m ap t:B is an iso­ m orphism for all B. (This is equivalent to saying that G is full and faithfui i.e. that it induces a bijection between m orphism s A^>B and m orphisms G,4-»-GB for each pair (A, B) - see [M ac Lane 1971], p. 88, Theorem 1.) If both t] and £ are isomorphisms, we call the adjunction an equivalence (cf. [M ac Lane 1971], p. 91), and say th at the categories C and D are equivalent. (We say C and D are dual if C is equivalent to the opposite category D°p of D.) The notion of equivalence is weaker than th at of isomorphism between categories (which may be regarded as an adjunction

Some categorical concepts

19

for which the unit and counit m aps are identities), but it is nonetheless sufficient to ensure that C and D share the same categorical properties, (in fact this last rem ark m ore or less defines w hat a categorical property is - cf. [F reyd 1976].) 3.5 We next introduce the im portant categorical concept of limit. Let C and J be categories (in what follows, J will invariably be small; th at is, its objects and m orphism s will form a set rather than a proper class). We may form the category C J whose objects are functors J -» C (also called, in this context, diagrams of type J in C) and whose m orphisms are natural transform ations; and we have a functor A : C - » C J which sends an object A to the constant functor with value A. We say C has limits of type J if A has a right adjoint lim j, and we refer to lim j(D) as the limit of the diagram D. (Dually, if A has a left adjoint we denote it lmij and say C has colimits of type J ; it is easy to see that the colimit of D : J -» C is just the limit of D : J op-»C°p.) We next investigate some particular cases of this notion (see also [M ac Lane 1971], pp. 62-71). Examples (a) Suppose J is the empty category 0. Then C J is the category 1 with one object and one (identity) m orphism , and lim j is a functor which picks out an object 1 of C such that every object of C has a unique m o r­ phism into 1. Such an object is called a terminal object (dually, initial object). {b) Let J be the category with two objects 1, 2 and two identity m o r­ phisms. Then C J is the cartesian product C x C . F or a pair of objects ( A u A 2\ l i mj ( ^ i , A 2) is usually denoted by A t x A 2; it comes equipped with a pair of m orphism s (pt : A x x A 2-> AU p2 : A i x A 2->A2) (the counit of the adjunction) such that for each pair ( / : A u g : B^>A2) there is a unique Al x A 2 satisfying p x(f, g ) = f and p 2( f g) = g. Such an object is called a (categorical) product of A t and A 2 in C. (c) M ore generally, let J be any discrete category (i.e. one with only identity morphisms). Then a functor D : J-> C is just a J-indexed family of objects of C (where J is the set of objects of J), and Um j (D) = n jeADJ) is the categorical product of the objects D(j). In a poset, products are the same things as meets; i.e. (d) Let J be the category represented diagram m atically by •=?*. A functor J ^ -C is a parallel pair of m orphism s of C (i.e. a pair with the same dom ain and codom ain); the limit of a parallel pair A z $ B is a m orphism e : E ^ A such that f e = ge, and such that if h :C->A satisfies fh = gh
idD. The ‘trace’ of this adjunction on the category C includes the com posite functor T = G F : C-+C, the natural transform ation q : id c -* ^ and the natural transform ation h = G ef : T T —GFGF-~+GF=T. In this context, the triangular identities yield the com m utativity of

and naturality ofe yields the com m utativity of TTT

Tfi

TT

We define a monad on C to be a triple T = (7^ rj, //) where T is a functor

Some categorical concepts

21

C -+C and r], \i are natural transform ations satisfying the above com ­ m utative diagrams. Given a m onad T on C, we can always find an adjunction inducing it. W e define a T -algebra in C to be a pair (A, a), where A is an object of C and a: TA-+A a m orphism such that

commute. A homomorphism of T-algebras / : (A, a)-*(B, /?) is a m orphism / : A - ^ B of C such that

commutes. W e thus have a category C T of T-algebras and hom om or­ phisms, and there is an obvious forgetful functor G : C T->C which sends (A , a) to A. But if A is any object of C, then (TA , fiA) is a T-algebra (the free algebra generated by A); this defines a functor F : C -* C T. It is then easy to verify that there is an adjunction F H G , which induces the original m onad T on C (see [M ac Lane 1971], p. 136, Theorem 1). N ow let Cs=±D be any other adjunction inducing the same m onad G' _ T on C. Then there is a comparison functor K : D -+C , which sends an object B to the T-algebra (G'B, G'(ei) : TG 'B = G'F'G'B-tG'B) (where e' is the counit of (F -(G ')). This functor is the unique functor satisfying G K = G ' and K F r= F(\_M ac Lane 1971], p. 138, Theorem 1); we say that the adjunction (F '-iG ') (or simply the functor G') is monadic if K is part of an equivalence of categories. It is often useful to know th at a particular functor is monadic, since the particular form of the category CT means that it inherits many good categorical properties (for example, existence of limits) from C, and these properties can then be transferred to D. 3.7 M any of the concrete categories with which we have to deal (for example, the last five on the list in 3.2) have the com m on property

22

I : Preliminaries

th at the ‘structure’ on the underlying sets of their objects consists of a num ber offinitary operations (i.e. functions from a finite cartesian power of the set to the set), which are required to satisfy a num ber of equational conditions (e.g. com m utative and distributive laws). Such categories are called finitary algebraic categories (an older nam e is ‘varieties of algebras’); they have a good many special properties which m ake them convenient categories in which to work. Proposition Let A be a finitary algebraic category. Then the forgetful functor G : A -*Set is m onadic (indeed, it is ‘strictly m onadic’; i.e. the com parison functor K : A -»SetT is not just an equivalence but an isomorphism). Proof Suppose for convenience th at the com m on nam e of the objects of A is ‘widget’. We construct the free widget generated by a set X by an obvious generalization of the construction of free groups (cf. [M ac Lane 1971], pp. 140-2): we first consider the set of all ‘polynomials’ or words obtained by (inductively) applying widget operations to the el­ ements of X, and then factor by the equivalence relation which identifies two words iff they can be proved to be equal using the widget equations. The resulting set has an obvious widget structure, and is the free widget generated by X. Thus we have a functor F :S et-» A left adjoint to G. (Note: this description makes the construction of F(X) sound easier than it actually is. In particular cases, the determ ination of the equivalence relation mentioned above may be a decidedly nontrivial problem (the word problem for widgets).) We now have a m onad T on Set induced by (b —IG); it remains to show th at every T-algebra structure on a set A is induced by a unique widget structure, and that T-algebra hom om orphism s are the same thing as widget hom om orphism s. Roughly, this works because if (A, a) is a Talgebra, then for every n-ary operation co and rc-tuple (ai , . . . , an) of elements of A, the word w(au . . . , an) determines an element of TA, and the value of a at this element is the value of a>A at the given rc-tuple. (For a formal proof, see [M ac Lane 1971], p. 152, Theorem 1.) □ 3,8 There is a partial converse to the last proposition. If T is any m onad on Set, we can describe its algebras by operations and equa­ tions, provided we are willing to allow infinitary operations, i.e. functions defined on an infinite cartesian power of the underlying set. W hat we do is to fix one set X of each possible cardinality, and regard the elements

Some categorical concepts

23

of T X as 'X -ary operations1, together with a list of equations which say th at these operations fit together in the right way. (For more details, see [M anes 1976], C hapter 1.) However, there is still a problem here. The presentation by operations and equations described above clearly has only a set of operations of any given arity; but there is no reason to suppose that we can put an upper bound on the arities of the operations we must use, and so we may have a proper class of operations altogether. If we had only a set of oper­ ations, we could still construct free widgets as in 3.7 (with a suitable m odi­ fication to allow infinitely long words); but in general we are going to have a proper class of words to deal with. In a particular case, we may be able to solve the word problem to the extent of showing that every word is equivalent to one in a particular set, and then we can proceed as before. (The second half of the proof of Proposition 3.7 works quite happily with a proper class of operations, provided we know th at free widgets exist.) We shall call a category algebraic if it is m onadic over Set, and equationally presentable if its objects can be described by (a proper class of) operations and equations; in the next section we shall meet examples of equationally presentable categories which are not algebraic, and of algebraic categories which cannot be presented using only a set of oper­ ations. The following proposition summarizes some of the im portant features of algebraic and equationally presentable categories. Proposition Let A be an equationally presentable category. Then (i) A has all small limits, and they are constructed exactly as in Set (i.e. the forgetful functor A -»Set preserves them). (ii) If the forgetful functor A ->Set has a left adjoint, then A is algebraic, and has all small colimits. (iii) The m onom orphism s in A are exactly the injective hom o­ morphisms. (iv) Epim orphism s in A need not be surjective; but the regular epis in A are exactly the surjective hom omorphisms. (v) Every m orphism in A can be factored (uniquely up to iso­ morphism) as a regular epim orphism followed by a m ono­ morphism. Proof See [M anes 1976], C hapter 1.

□

24

I : Preliminaries |3.9 A particular class of colim its, which will play an im portant role in

chapter 6, is that of colim its over filtered categories. W e say a (small) category J is

filtered if (i) J is nonempty. (ii) Given any two objects j , f of J , we can find a third object f and m o r­ phism s in J. (iii) Given a parallel pair of m orphism s j r j / in J , we can find a third m or-

$

phism y : / - * / ' such th at ya = yfi. N o te that conditions (i)—(iii) are all special cases of the assertion "Given a finite diagram in J , we can find a cone under if (for (i), take the em pty diagram ); and in fact they are sufficient to imply the general form of the assertion (in m uch the same way as the existence of finite coproducts and coequalizers implies the existence of all finite colimits). The significance of filtered colim its ( —colim its over filtered categories) is en­ shrined in th e following well-known result: T heorem In Set, filtered colim its com m ute with finite limits; i.e., given a doublyindexed diagram D : I x J-+ S e t where I is finite and J is filtered, the canonical m ap

lira j Usi] D—^Umi liraj ^ is a bijection. Proof See [M ac Lane 1971], p. 211. (Here lirQj lilHi D denotes the colim it of the functor J-+ S e t obtained by regarding D as a functor I-+ S e tJ and taking its limit in S e tJ; similarly for lirQj liiUjZX) □ Corollary Let A be a finitary algebraic category. Then (i) T he forgetful functor G : A -+Set creates filtered colimits. (ii) In A, filtered colim its com m ute w ith finite limits. Proof (i) Given a diagram D : J-+ A , where J is filtered, let L be the colimit in Set of GD. Then for each natural num ber n the cartesian power L" is the colim it of the diagram (/V*GD(j)n); so we deduce th at there is a canonical way of im posing an algebra structure on L, which m akes it into the colim it of D in A. (ii) follows from (i), the Theorem and the fact that G also creates limits. □ Since the free functor F : S et-»A preserves all colimits, we deduce that if A is a finitary algebraic category, then the functor p art T = GF o f the m onad on Set which defines it m ust preserve filtered colimits. In fact this property characterizes those m onads on Set which correspond to finitary algebraic categories; for this reason, a functor which preserves filtered colim its is often said to be finitary. Lim its over filtered categories are not in general interesting; but in section VI 2

Free lattices

25

we shall devote some attention to cofiltered limits (i.e. limits over categories J such th at J op is filtered). Exercise Let J denote the poset of natural num bers with the opposite of their usual ordering, I the category represented diagram m atically by •=£-. By considering the I x J-indexed diagram in Set

where N is the set of natural num bers and s is the successor m ap (i.e. s(n)—n + 1 ), show th at the dual of the Theorem does not hold in S et (i.e. filtered colimits do not com m ute with finite limits in Set°p).

4. Free lattices 4.1 In this section, our main aim is to investigate word problems (in the sense of paragraph 3.7) for various categories of lattices; but we begin with some rem arks about lattices which are complete (and cocomplete) in the categorical sense, i.e. they have meets and joins for arbi­ trary subsets and not just finite ones. There is one special class of joins which we shall wish to consider. A poset A is said to be (upwards) directed if (i) A is nonem pty and (ii) every pair of elements of A has an upper bound (not necessarily a least upper bound) in A. We say A has directed joins if V S exists for every subset S ^ A which is directed in the induced ordering. (In the terms of paragraph 3.9, directed joins are just filtered colimits.) Lemma If a poset A has finite joins and directed joins, then it has arbitrary joins. Proof Let S £>4, and consider the set T of all joins of finite subsets of S. It is clear that T is directed (indeed, it is a sub-join-semilattice of A \ and that a e A is an upper bound for T iff it is an upper bound for S; so VS=VT.

□

26

I: Preliminaries

4.2 Next we present the A djoint F unctor Theorem. This is actually a general theorem of category theory, but its proof for 'large’ categories is hedged around with technical com plications called 'solution-set con­ ditions’ ; since we shall require the theorem only for m aps between posets (in which case the solution-set condition is autom atically verified), we shall state and prove it only in this special case. Theorem Let g : A-+B be an order-preserving m ap between posets. Then (i) If g has a left adjoint / : B--+A, g preserves all meets which exist in A. (ii) If A has all meets and g preserves them, g has a left adjoint.

Proof (i) Let S be a subset of A such th at A S exists. Since g is order-preserving, g( A S ) is clearly a lower bound for {g(s)\s e S}. But if b is any lower bound for this set, then we have b^g{s) for all s e S, whence f ( b ) ^ s for all s e S, so f ( b ) ^ A S and b ^ g ( AS). (ii) By the definition of an adjoint, f ( b) m ust be the smallest a e A satisfying g(a)^b. So consider f { b ) = A { a e A\ g{a)^b}.

Since g preserves meets, we have gf(b) = / \ { g ( a ) \ g ( a ) ^ b } ^ b \

and f g( a) = A{a' \g{a')^g(a)}

a

since a e {a'\g(a')^g(a)}. W e can regard these inequalities as natural transform ations idB-* g f fg-+ idA; and they trivially satisfy the triangular identities since A and B are posets. So / is left adjoint to g. □

Exercise O bserve that in a Heyting algebra the operation a-+(~) is by definition right adjoint to a a ( - ) . Give new proofs of Lemma 1.10(iv), Lem m a 1.1 l(i) and Exercise 1.11 (i) using the A djoint F unctor Theorem.

4.3 A join-sem ilattice A is said to be complete if it has arbitrary joins and not just finite ones. The next result is actually a corollary of Theorem 4.2 [ Exercise: why?], but we give an independent proof.

Free lattices

27

Proposition A poset is a complete join-sem ilattice iff it is a complete meetsemilattice. Proof Let A be a complete meet-semilattice, C onsider the set T of upper bounds for S, and let a — A T Since every s e S is a lower bound for T, we have s ^ a and hence a is an upper bound for S. So a is the least element of % i.e. a — V S. □ Thus every complete semilattice is actually a com plete lattice; but it is convenient to distinguish between the two concepts, since a hom om or­ phism of complete (join-)semilattices (i.e. a m ap preserving arbitrary joins) is not necessarily a hom om orphism of com plete lattices (i.e. it need not preserve meets). We thus have tw o categories C S L at and C L at with the same objects but (as we shall see) very different categorical structure. Exercise Show that the category C S L at is isom orphic to its opposite. [C onsider the functor which sends a com plete semilattice to its opposite, and a hom om orphism to its right adjoint.] 4.4 It is clear that the theory of com plete semilattices can be given an equational presentation, by adjoining to the presentation for semi­ lattices (1.3) an rc-ary operation supn for each infinite cardinal n, and equations which say that supn ( / : n-*A) does indeed give the join of the image of f (A suitable set of equations is given in [M anes 1976], p. 69.) Similarly for complete lattices, etc. However, these presentations involve a proper class of operations, so we cannot immediately conclude that the corresponding categories are algebraic. In the rest of this section, we shall investigate the existence and nature of free objects in C SLat, C L at and the category CBool of complete Boolean algebras, as well as the finitary algebraic categories SLat, Lat, D L at and Bool. We begin with the simplest case, that of semilattices. Lemma The free semilattice generated by a set X is the set F X of finite subsets of X , the semilattice operation being union. Proof The unit m ap rjx : X - + F X sends x e X to the singleton set {x}. N ow any set S e F X is uniquely expressible as a finite union of single­

28

I: Preliminaries

tons, so any m ap / : X- * A, where A is a semilattice, has a unique extension to a semilattice hom om orphism f : F X ^ A ; S » V A{f(x)\xeS}.

□

N ote in particular that the free semilattice generated by a finite set is finite. (This observation is not quite as banal as it seems; it is related to the definition of finiteness in axiom atic set theory - see [K uratow ski 1920].) 4.5 no surprises:

The extension of Lemm a 4.4 to com plete semilattices holds

Lemma The free complete semilattice generated by a set X is the powerset P X , the semilattice operation being union. Proof Exactly like Lemma 4.4.

□

Corollary The category C SL at is algebraic.

□

t4.6 T he w ord problem for free lattices is less trivial (and m ore interesting). If X is a set of generators, let us write W X for the set o f all words constructed from X using th e lattice operations. W e define (inductively) a binary relation ^ on W X as follows: p ^ q holds iff one of the following seven conditions is satisfied: (1) p and q are the same individual generator. (2 ) p is the constant 0. (3) q is the constant I. (4) p=(pi v p 2) and both p ^-ideaP (a and fi standing for addition and m ultiplication) for w hat we now call ‘ideal’ and ‘filter', but these term s did not catch on.) T he first explicit appearance of the M axim al Ideal Theorem (Lemm a 2.3) seems to have been in [Stone 1936], though the idea was present in [U lam 1929], [Tarski 1930] and [Birkhoff 1933]. As we rem arked in the text, the M axim al Ideal Theorem is closely related to the axiom of choice; let us denote by ‘M IT for X ’ the statem ent ‘Every nontrivial X has a maximal ideal’, and by ‘P IT for X ’ the same statem ent with ‘prim e’ in place o f ‘m aximal’. Then we showed in the text th at the axiom of choice (in the form of Zorn’s Lemma) implies M IT for arbitrary lattices (Lemma 2.3), and th at M IT implies P IT for distributive lattices (Corollary 2.4). It was shown by Scott [1954], M row ka [1956, 1958] and Klim ovsky [1958] th at the M IT for distributive lattices (indeed, for Heyting algebras) is equivalent to the axiom of choice. O n the other hand, for Boolean algebras the M IT is equivalent to the PIT, by Proposition 2.6; and it can be shown [Scott 1954a] th at the P IT for Boolean algebras is sufficient

N o te s on chapter I

37

to imply P IT for arbitrary distributive lattices (and for arbitrary com m utative rings as well). F o r some tim e it was an open problem w hether P IT was equivalent to the axiom of choice; this problem was settled negatively by H alpern [1964], who ex­ hibited a m odel of set theory in which the axiom of choice fails but P IT holds. (M ore recently, Hodges [1979] has shown th at the M IT for rings (indeed, for unique factorization dom ains) is sufficient to imply the axiom of choice.) Although the P IT is thus weaker than the axiom of choice, it is certainly not provable in Z F set theory; and it can be shown to imply a weak form of the axiom of choice, nam ely that every family of nonem pty f inite sets has a choice function. In subsequent chapters, we shall see th at m any ‘non-constructive’ theorem s in general topology and elsewhere are actually equivalent to the PIT. The analogue for com m utative rings of Theorem 2.4 (in which U lter’ is replaced by ‘m ultiplicatively closed subset’) was first proved by K rull [1928]; the passage from there to the theorem for distributive lattices was a mere formality. (F or an interesting sidelight on the logical status of Theorem 2.4, see [Johnstone 1979c].) P roposition 2.5, and the representation theorem which follows it as Exercise 2.5, are due to Birkhoff [1933]; as m entioned in the Introduction, this is simply Stone’s representation theorem w ithout the topology.

Section 3 The concepts of category, functor and natural transform ation (3.1) were introduced by Eilenberg and M ac Lane [1942, 1945]. A djunctions (3.4) m ade their first explicit appearance in [K an 1958], though the idea had been ‘in the air’ for som e time before then, notably in B ourbaki’s [1948] notion o f ‘universal construction’ (see also [Sam uel 1948]). The categorical notion o f limit (3.5) has its origins in M ac Lane's [1948, 1950] observation th at cartesian products could be described by universal properties; other particular examples of limits (notably filtered colim its (3.9)) were studied by m any people, but the general concept was first form ulated by K an [1958]. M onads (3.6) first appeared in [G odem ent 1958], under the som ew hat uninspired nam e of ‘standard constructions’; they were known for som e years by the equally uninspired nam e ‘triples’, and the popularization of their present nam e is largely the w ork of M ac Lane. The theory o f m onads becam e of prim ary im portance with the construction by Eilenberg and M oore [1965] of the category of algebras for a m onad. Finitary algebraic categories, under their older nam e ‘varieties o f algebras', were first studied by Birkhoff [1935], w ho proved th at free algebras exist (Proposition 3.7). A m ore categorical form ulation of the sam e notion, taking the concept of free algebra as basic, is due to Lawvere [ 1963]; its extension to infinitary operations is the w ork of L inton [ 1966,1969], who also determ ined the relationship between equation­ ally presentable and algebraic ( = m onadic) categories. F o r a m ore detailed account of this m aterial, see [M anes 1976] or [W raith 1969].

Section 4 T he A djoint F unctor Theorem , in its general categorical form, is due to Freyd [1964]; the lattice-theoretic version which we give as Theorem 4.2

38

I : Preliminaries

has been discovered independently by m ore authors than anyone would care to count, as also has Proposition 4.3. Exercise 4.3 is another folk-theorem, which seems to have m ade its first explicit appearance in [Johnstone 1978]. Lem m a 4.5 is due to M anes [1967] (but see also [Schm idt 1957]); it is interesting to note that both this and Lem m a 4.4 remain true under m uch weaker assum ptions than those of Z F set theory (see M ikkelsen 1976]). The w ord problem for free lattices (Lemma 4.6) was solved by W hitm an [1941, 1942]. Its extension to com plete lattices, and P roposition 4.7, are the work of Hales [1964]. The structure of free distributive lattices (4.8) was known to Skolem [1913], and in principle even to Dedekind [1897], who first posed the problem of deter­ m ining the num ber of elements in the free distributive lattice on n generators. F or recent w ork on the latter problem, see [M arkow sky 1973, 1980]. (Although the theory of m odular lattices will not concern us at all, it is perhaps appropriate to note here that Freese [1980] has recently settled a long-standing problem by showing that the word problem for free m odular lattices is recursively insoluble.) P roposition 4.10 was proved independently by G aifm an [1964] and Hales [1964], both of whom used a ‘word-problem ' approach like th at which we used in Proposition 4.7; the ‘m odel-theoretic’ argum ent which we use to prove 4.10 is due to Solovay [1966]. Free Heyting algebras (4.11) have been studied by N ishim ura [I960], U rquhart [1973] and Freyd [1984].

II Introduction to locales

1. Frames and locales 1.1 In this chapter we em bark on the study of topological spaces in terms of their open-set lattices. If X is a topological space, the lattice Q(X) of open subsets of X is complete, since an arbitrary union of open sets is open; and the infinite distributive law a

a

V S = V {a a s \ s e S}

holds in Q(X), since it holds in the Boolean algebra P X and the inclusion Q ( X ) - * P X preserves finite meets and all joins. By the Adjoint Functor Theorem (I 4.2), a complete lattice satisfies the infinite distributive law iff it is a Heyting algebra; however, the natural m orphism s to consider between lattices of the form Q(X) are not Heyting algebra hom om orphism s (they do not preserve implication), and so in keeping with the policy outlined in I 3.2 we shall introduce a new name (in fact, two new names) for these lattices shortly. If f : X - ~ + Y is any map, then / 1 is a hom om orphism of complete Boolean algebras from P Y to P X ; and if / is continuous, then f ~ l restricts to a m ap Q(Y)-+Q(X), which clearly preserves a and V. This motivates the following definitions: Definition (a) The category Frm of fram es is the category whose objects are complete lattices satisfying the infinite distributive law, and whose m orphisms are functions preserving finite meets and arbitrary joins. (b ) The category Loc of locales is the opposite of the category Frm. We refer to m orphisms in Loc as continuous maps , and write Q for the 39

40

I I : Introduction to locales

functor Sp-+Loc which sends a space to its lattice of open sets, and a continuous m ap f : X - + Y to the function : Q(Y)-»Q(X). Thus so long as we are concerned only with objects , the terms ‘frame’, lo cale’ and ‘complete Heyting algebra’ are entirely synonymous; it is only when we refer (expressly or implicitly) to m orphism s th at they become different. For example, a subframe of a frame A is simply a subset of A which is closed under a and V ; but a sublocale is something different, corresponding to a quotient frame. (We shall encounter the precise definition of a sublocale in the next section.) O f course, if / : A-+ B is a frame hom om orphism , it has a right adjoint g which goes in the ‘localic’ direction by Theorem I 4.2; but there does not seem to be any condition on g alone which is equivalent to / preserving finite meets. Exercise Let f : A - > B and g : B-> A be m aps between complete Heyting algebras with f - \ g . Show that / preserves binary meets iff the equation g(fa-+b) = (a-+gb)

holds for all a e A, b e B. C an you find a similar condition for / to pre­ serve 1 ? We adopt the convention that if f : A - * B is a continuous m ap of locales, we shall write / * : B-+A for the corresponding frame hom o­ morphism, and /* : A-+ B for the right adjoint of / * .

1.2 T heorem The category Frm is algebraic. P roof Frm is clearly equationally presentable, so it suffices to construct a free functor Set-* Frm. In fact we shall construct a left adjoint for the forgetful functor Frm -»SL at which sends a frame to its underlying meet-semilattice; since we already know how to construct free semi­ lattices (I 4.4), this is sufficient. Let A be a meet-semilattice. Let DA denote the set of all lower subsets of A , and let q : A- +DA send a to j(a). If we order DA as a subset of P A , it is clearly a sub-complete-lattice of PA; so the infinite distributive law (1.1) holds. (So does its dual, though this a mere coincidence.) M oreover, by the defining property of meets, we have

41

Frames and locales

i(a)nl (b) = i (aAb)

and 1(\ a) = A; s o r] is a semilattice hom omorphism. Now let / : be a meet-semilattice hom om orphism , where B is a frame. Define / : D A-+B by 7 (S )= V { /(s )|s e S ); B

it is immediately clear that f is order-preserving, and it extends / since f ( a) e {/(s)|s e |(a)} £ j( /( a ) )

implies f ( l ( a) ) =f ( a) . It is also clear from the form of the definition that f preserves joins, so it remains to show that it preserves finite meets. But AS)Af{T) = (y{f(s)\seS})A(y{f(t)\teT})

= V { /( s ) a /( 0 |s e S , t e T } by the infinite distribu­ tive law in B —V

( / (

s a

t)\s e S , ( e T } since / preserves

a

= V { / (u)\u e S n T } since S and T are lower sets =f(SnT).

The uniqueness of / is obvious from the fact that for any S e DA we have V da {!(a)|a e S}, and this join must be preserved by / □ 1.3 Given a locale, can we find a space which ‘best approxim ates’ it? Since a point of a space X is the same thing as a continuous m ap 1 where 1 is the one-point space, it seems reasonable to define a point of a locale A to be a continuous m ap Q(l) = 2-»,4, i.e. a frame hom om orphism p : 2. By I 2.2, such a m ap is completely determined by its kernel 1(0) or its dual kernel p~ ^ l), which are respectively a prime ideal and a prime filter of A. But since p preserves arbitrary joins, it is clear that p ~ 1(0) m ust be a principal ideal, since we have p( V(/?“ 1(0))) = 0 and so p ~ l(0) = l ( \ / ( p ~ 1(0))). (Equivalently, r( l ) must be a completely prime filter, i.e. one satisfying VS

(3s e5)(s e p ~ l (l)).)

Thus points of A correspond bijectively to prime elements of A, i.e. elements generating prime principal ideals. We write pt(A) for the set of points of A. Now let a e A. Define (a) to be the set of points p : A-+2 such that p(a) = 1 (or equivalently the set of prime elements x e A with x ^ a ) . Lemma (j) is a frame hom om orphism A->P(pt(A))\ in particular its image is a topology on pt(A).

42

I I : Introduction to locales

P roof Let p be a point of A, S a subset of A. Then P € [J{(t>{a)\a € S}(3a € S)(p(a)= 1)

V {p(a)\a € S} = 1 /?( V S) — 1 since p preserves joins /? g ( V S), so (f> preserves joins. The proof that preserves finite meets is similar. □ H enceforth we shall always consider pt(^4) as a topological space, with topology given by the image of ; and we shall regard itself as a con­ tinuous m ap Q(pt(yl))-^A in Loc.

1.4 Theorem The assignment ,4i->pt(,4) defines a functor Loc-»Sp, which is

right adjoint to Q. P roof It suffices to prove that any continuous m ap / :Q(X)-+A factors uniquely through :Q(pt(A))-+A by a m ap / :X ->pt(/4) in Sp (This will in particular prove that pt is a functor.) Suppose given such an f For each x e X , the composite 0 ( 1) - ^

Q (X ) -± -+ A

is a point of A, which we denote /(x ). So we have a m ap / : X-^pt(yl); and / is continuous, since for any a e A we have r

, ((a))={xeX\n(x)*(f*(a))=l) = {x e X \ x ef *( a) } =f * ( a)

which is open in X . This argum ent also shows that (j) *Q ( / ) = / in Loc; the fact that this equation determines / uniquely is obvious. □

1.5 Example In a Boolean algebra, it is not hard to verify that an element is prime iff its complement is an atom. Thus if A is a complete Boolean algebra, considered as a locale, we may identify the points of A with its atom s; and (under this identification) the m ap sends a e A to the set of atom s x such that x ^ a . In particular, (j) is one-to-one iff A is atomic. In general, therefore, the m ap (j) : A->Q(pt(A)) is not an isomorphism

43

Frames and locales

of frames; for a general A , it is iso iff (Va, b € A)(a^b=^(3/? € pt(A))(p{a) = 1 and

p(fc) = 0)).

A locale satisfying this condition is called spatial , or said to /iaue enough points. Clearly, if X is any space then Q(X) is a spatial locale, since in the formula above we can take p to be Q(x) for any x € X which is in the open set a but not in b . Exercise Show that a locale A is spatial iff each element of A can be ex­ pressed as a meet of prime elements. Show also th at if A and B are spatial locales and / * : B- +A is left adjoint to /* : A-+B, then / * preserves finite meets (i.e. is a frame hom om orphism ) iff /* preserves prime elements.

1.6 If X is a space, how does it com pare with pt(Q(X))? The points of Q(X) correspond to prime elements in the lattice of open sets of X ; their complements are the irreducible closed subsets of X , i.e. those closed F £ X which cannot be written as a union F = F 1u F 2 where both F x and F 2 are proper closed subsets of F. For any x € X, the closure of {x} is irreducible, since if we express it as a union of closed subsets one of them must contain x; this defines a m ap ij/ : X-»pt(Q(X)), which is in fact the unit of the adjunction of Theorem 1.4. We say X is sober if i/^ is a bijection, i.e. if every irreducible closed subset of X is the closure of a unique point of X. N ote that ij/ is then autom atically a hom eom orphism, since Q(ij/) is the isomorphism : £l(X)->n(pt(£l(X))). Lemma (i) A space X is T0 iff : X -►pt(Q(X)) is one-to-one; in particular, any sober space is T0. (ii) Any H ausdorff space is sober.

P roo f (i) If x, y are points of X, then we have ^(x) = ij/(y) iff x and y are contained in exactly the same open sets of X, iff the T0 axiom fails for the pair (x, y).

(ii) If F is a closed subset of a H ausdorff space X containing two distinct points x and y, let U and V be disjoint open neighbourhoods of x and y. Then F — U and F — V are proper closed subsets of F whose union is F —( U n V ) = F , i.e. F is reducible. So the only irreducible closed subsets of X are singletons, i.e. ij/ is bijective. □

44

II: Introduction to locales

However, there is no implication in either direction between sobriety and the 7~i axiom. F o r example, an infinite set X with the cofinite topology is 7"i but not sober; X itself is an irreducible closed subset, and pt(Q(X)) is sober but not Tl .

1.7 Lemma For any locale A , the space pt {A) is sober. P roo f Let F be an irreducible closed subset of pt(,4). Its com p­ lement in pt(A) is a prime open set, which we can write as {a) or <j)(c) ^(f){a) and so b ^ a or c ^ a . Moreover, it is easy to see that the point p of A defined by this prime element satisfies tj/(p) = pt (A) —(j)(a) — F. So tjf is surjective. But if /?, q are distinct points of A, then there exists a e A with p(a)=/=q{a\ so that the open set (j)(a) contains just one of p and q ; thus pt (A) is a T0-space. So by Lemm a 1.6(i) ij/ is injective. □

Combining the above Lemma with the results of 1.4 and 1.5, we obtain Corollary (i) The adjunction of Theorem 1.4 restricts to an equivalence of categories between the (full) subcategory S o b ^ S p of sober spaces and the subcategory S L o c ^ L o c of spatial locales. (ii) The inclusion Sob-»Sp has a left adjoint, namely the com­ posite pt *Cl (pt(Q(X)) is called the soberification of X.) (iii) The inclusion SLoc-»Loc has a right adjoint, namely the composite Q • pt. □ Exercise Show that the following conditions on a sober space X are

equivalent: (i) For every x e X, {x} is open in cljxj. (A space satisfying this condition is called a TD-space.) (ii) There is no proper subspace of X whose s o b e r ificat ion is (homeomorphic to) the whole of X.

Frames and locales

45

1.8 We may define a partial order on the points of any T0-space X by x ^ y iff x is in the closure of {y} (equivalently, cl{x} £cl{y}). If this relation holds we say x is a specialization of y. It is clear that the relation ^ is reflexive and transitive; its antisym m etry is precisely the T0 axiom. N ote also that any continuous m ap between 70-spaces is necessarily order-preserving, and that the order is discrete (i.e. satisfies x ^ y iff x = y) iff X is a -space. In the converse direction, suppose we are given a poset (X, ^ ). C an we find a T0 topology on X for which ^ is the specialization ordering? In fact there are two canonical answers to this question, one maximal and the other minimal. We define the Alexandrov topology Y(X, (or simply Y(X), if the partial order is obvious) to be the collection of all upper sets in X (i.e. sets U such that x € U and x ^ y imply y € l / ) ; this is clearly a topology, since it is closed under arbitrary unions and intersections. And we define the upper interval topology <J>(X, ^ ) (sometimes called the weak topology) to be the smallest topology for which all sets of the form j(x) are closed, i.e. the topology based by sets of the form X - ( i ( X ! ) u ••• uj,(x„)). Proposition Let (X, be a poset, Q a topology on X. Then Q (is T0 and) induces the ordering ^ iffO(X, ^ ) ^ Q ^ Y ( X , ^ ). Proof Suppose Q induces Then every Q-open set must be an upper set, since if cl{y} = j(y) meets an open set U then y e U, and so Q ^ Y . But since J,(y) must be Q-closed for every y, we also have O Conversely, if £Y , then it is easily seen that j(y) is the smallest Qclosed set containing y, and so x ^ y iff x ecl{y}; and Q is T0, since it contains the T0 topology O. □ Exercise Let (X, Q) be a T0-space. Show that the following conditions are equivalent: (i) Q coincides with the Alexandrov topology for the partial order it induces. (ii) Q is closed under arbitrary intersections. (iii) Every point of X has a smallest open neighbourhood. tl.9 F o r sober spaces, the relation between topologies and partial orders is less simple. First of all, sobriety implies a nontrivial property of the induced o rd e r:

46

I I : Introduction to locales Lemma

If (X, Q) is a sober space, then the specialization ordering on X has directed joins. M oreover, if U eQ , then V is not merely an upper set but also ‘inaccessible by directed joins’, i.e. S ^ X directed and V S e V imply S r \ U i = 0 . Proof Let S be a directed subset of X . Then the family of subsets

{cl{xj|x s S ] is also directed; let 7 be the closure of its union. If T = F l u F 2 where F t and F 2 are closed, then for each x e S w e have either x e f j or x e F 2, and by directedness we conclude that either S ^ F ^ or S ^ F 2, since the F, are lower sets. Hence T = F t or T = F 2, i.e. T is irreducible. So T is the closure of a unique point y\ which is readily seen to be the join of S in (X, ^ ) . M oreover if V is open and y e U, then U meets ( J ic l{ x ] ]x e S j, which implies that x e U for some x e S . So U is inaccessible by directed joins.

□

If (X , is a poset with directed joins, then the set 2 (X, of upper sets which are inaccessible by directed joins is easily seen to be closed under finite intersections as well as arbitrary unions; so it is a topology on X, called the Scott topology. (Note that the sets X — j(x) are inaccessible by directed joins - indeed by all joins which may exist in X - and so (A', ^ ) ^ E ( A ', ^ ).) F o r sober topologies, we may thus improve the bounds of P roposition 1.8: a sober topology Q on X induces a given partial order ^ iff (tyX, ^ ) . O n the other hand, there is no reason in general to suppose that either O or I is sober, or even that there is any sober topology lying between them : Exercise

Let X be the set N x (

j oc j ), with partial order defined by

(m, n ) ^ { m \ n')k is clearly order-preserving; so con­ dition 2.2(i) is satisfied. To verify condition 2.2(iii), note first that i f b ^ a then (j(b)->k(b))^(j-*k){a), and so j( j( b) ^ k{ b )) ^k ( j( b )^ >k (b ) )^ ( j^ k )( j ^>k ) (a ) .

But (j(b)-*k(b)) e A k by Exercise 2.3(ii), so j(j(b)^k(b))^k(j(b)^m =j(j(b)^k(b))^(j(b)^k(b)) = (j(J{b)^k{b)) a j ( b ) ) ^ k ( b ) =j ( ( j( b) ^k (b )) * b ) ^ k { b ) =j(b)^k(b)

since b^k{b)^:j{b)->k{b). But this is true for any b ^ a , so ( j ^ k ) ( j ^ k ) ( a ) ^ : A {j(b)^>k(b)\b^a} = O'-^Xa).

Thus ( j—*k) 6 N{A). N ow for any a e A we have j ( a ) a ( j ^ k ) { a ) ^ j ( a ) a ( j( a)^>k(aj)^k(a\

i.e.; a ( j ^ > k ) ^ k in N(A). But if / is any nucleus with j a /^/c, then for any b ^ a we have j(b) A l ( b ) ^ k ( b ) and so l(a)^l(b)^j(b)->k(b). Hence /(a )^ (jr—>fc)(a). □

t!6 Lemma

(i) F o r any a e A, the open and closed nuclei u{a), c(a) are com plem entary elements of N(A). (ii) The m ap a\-*-c(a) is a frame m onom orphism A ^ - N ( A ) . P r o o f (i) F or any b, we have (a v b) a ( a ^ b ) = (a a (a-*b)) v (b a (a->b))

= (a a b) v b =b

by I 1.10(ii, iii)

Sublocales and sites

53

so that c{a) Au{a) is the identity map, which is clearly the minimal element of N(A). N ow if b e A c{a)s/Uia)= A c{a)n A u(a), then a v b = b , so a ^ b and a - >b = 1; but a - * b = b , so b = 1. T hus c(a)vu(a) is the constant function with value 1^, i.e. the maximal elem ent of N(A). (ii) T he identity c ( f l A f c ) = c ( a ) A c ( b ) is a simple application of the distributive law. F o r a subset S £ A , c( V S ) is the m ap b h > V {c ( a) ( b) \ ae S j;

since this is a nucleus, it is clearly the join of the c(a), a e S, in N(A). Finally, c(a)=c(b) implies a = c(aXO) = c(i>)(0) —b, i.e. c is a m onom orphism .

□

Corollary

Every frame is isom orphic to a subfram e of a com plete Boolean algebra. P ro o f Given a frame A , let B = (N(A))-\-\. Then B is a com plete Boolean algebra by 2.2 and I 1.13; and by the Lem m a the m onom orphism c : A - * N ( A ) factors through B. (It is clearly still a frame hom om orphism when we consider it as taking values in B.) □

Although the construction of B from A in the C orollary is fairly canonical, it is not in general the free com plete Boolean algebra generated by A. The next four paragraphs have as their goal the construction of such a free algebra, when it exists. t!7 Proposition

(i) Every nucleus on A is expressible as a join of nuclei of the form c(a) a u(b). (ii) c : A - >N ( A) is an epim orphism in Frm. P r o o f (i) Let j be a nucleus on A, a s A. C onsider the nucleus k a = c(j{a )) a u{a). W e have k Ja) = {Ha) vfl) a (a->a)=j(a) a 1 =j(a).

But for any b , we have j(fl) a (a-> b)^/(a) a j{a-*b) = j(flA M ))

so th at a - *b^ j( a) - *j ( b ) . Hence 0(a) v b)->j{b) = ( j(a)^j(b)) a (b^>j(b))

= (/(a)-V(*>)) A 1 Since b ^ j ( b ) ^a^b.

54

I I : Introduction to locales

so that kJb)=U(a) vb ) a {a-*b)^j{b\ i.e. ka^ j in N(A). So j is an upper bound for {ka\a e A } in N(A). But any upper bound / for this set m ust have l ( a ) ^ k a(a)=j(a) for all a, i.e. l ^ j . Thus j — V {c(j(a)) a u(a)|a e A}. (ii) W e have to show that a frame hom om orphism / : N(A)-* B is determ ined by its effect on nuclei o f the form c(a). But if we kn ow f(c(a)) then we know f(u(a)), since c(a) and u(a) are com plem entary in N(A) and com plem ents in B are unique. Hence also we know f(c(a)Au(b)) for any pair (a , b), and so by the first part we know f ( j ) for any

since / preserves finite m eets and all joins.

□

f2 .8 Let / : B - > A be a continuous m ap o f locales, and j a nucleus on A. In any category, a pullback o f a regular m onom orphism is a regular m onom orphism , and so the pullback o f (Aj->A) along / is a sublocale o f B, which we can write as (Bf{j)-*B). W e thus have a m ap / : N ( A ) - * N ( B ) ‘, we first investigate its effect on open and closed nuclei. Lemma Let / : B->A be a continuous map o f locales, a e A. Then (i) / factors through A c(a)- ^A iff f * {a ) = 0. (ii) / factors through A u(a)->A iff f * (a ) = 1. (iii) For any £ f(c(a)) = c(f*(a)) and f(u(a)) = u(f*(a)). P ro of (i) Clearly, /

factors through A c{a)-+A iff / * factors through the

inclusion m ap A c{a)-*A. But since -4c(fl)= T(tf), this happens iff /* (0 B)^B which factors through Bc{f*ia)). So the universal such diagram (i.e. the pullback) has C = B cif*ia)). The argument for open nuclei is similar.

□

Sublocales and sites

55

Proposition The assignment A ^N {A \

/* h > /

defines a functor N : Frm -* Frm, and c : A - * N ( A ) defines a natural transform ation from the identity functor to N. P r o o f Since pullback squares can be com posed, / is clearly functorial in / * , and part (iii) o f the Lem m a show s that c is a natural transform ation. Se we have only to show that / is a frame hom om orphism . N o w / clearly preserves jo in s o f nuclei, since these correspond to intersections of sublocales and the latter (being lim its in

Loc) are preserved by pullback along a

fixed m orphism . So we need only consider finite m eets. Since 1

1. Next we consider m eets o f the form c(a)

a u (b)\

= c( 1^), / preserves

we have

f(c(a) a u(b)) v f(u(a) v c(b)) = f { 1) = 1 and f ( c { a ) A u ( b ) ) A f { u ( a ) v c ( b ) ) ^f(c{a))Af(u(b)) =f(c(a))

a

a

f ( u ( a) v c ( b) )

f ( u ( b )) a ( f(u(a)) v f(c(b)))

=0 since / preserves the com plem entarity o f c(a) and u(a), by the Lemma. Hence f ( c ( a ) a u(b)) = c ( f * ( a )) a u{ f* (b ) l since both are com plem ents in N(B) for f (u(a) v c(b)). W e note also that the meet of tw o nuclei o f the form c{a) a u(b) is again o f this form, since c(a} ) Ac{a2) = c(al a a2) and u(bx) Au(b2) = u(bl v b 2)', and it follow s from the above that /

preserves the

m eets o f pairs o f such nuclei. N ow let j, k be an arbitrary pair o f nuclei on A. U sing P roposition 2.7(i), we can write j = V S , k = V T where S, T are sets o f nuclei o f the form c(a)

a

u(b). N ow

/O ' a k ) = f ( V S a V T ) = / ( V {s a t\s e S, t e T })

since N ( A) is a frame

—V {f(s

since / preserves V

a

t)|s 6 S, t e T}

= V { / (s) a / (t)|s e S, t e T ]

by the above argument

= V{/(s)|seS}A V{/(r)|reT}

= / ( V S ) a/ ( V T ) = /0 ')A /( f c ) -

□

t2 .9 W e have seen that c : A - * N ( A ) is both m ono and epi in Frm. However, Proposition The m ap c : A - * N ( A ) is an isom orphism iff A is a Boolean algebra.

56

I I : Introduction to locales P r o o f Suppose A is Boolean. Then any sublocale o f A is Boolean, for if

a and b are com plem entary elem ents in A then j(a) and j(b) are com plem entary in Aj. So by Exercise 2.4(ii) every sublocale o f A has the form (v4c(0)) - n for som e a e A. But A c(a) is Boolean and so equals C4c(a)) - n ; hence every sublocale o f A is closed. Conversely, suppose every nucleus on A is closed. Then for every a t A, u{a) is closed. But if u(a) = c(b\ then a v b = c(b)(a) = u(a)(a) = a - * a = 1 and a a b = a Ac(b)(0) = a Au(a)(Q) = a a ~~la = 0,

□

so b is the com plem ent o f a. Corollary

Any frame hom om orphism / * \ A - > B , where B is a com plete Boolean algebra, factors uniquely through c : A - * N ( A ) . P ro o f U niqueness o f the factorization com es from Proposition 2.7(ii); existence from the diagram

N(B)

N( A)

□

and the last two Propositions.

|2.10

N ote that any frame hom om orphism between com plete Boolean

algebras preserves infinite m eets as well as joins, since it preserves com plem ents; that is, CBool is a full subcategory o f Frm. In the situation o f Corollary 2.9, we might reasonably expect to obtain a left adjoint for the inclusion CBool-*Frm by iterating the functor N a sufficient num ber o f tim es - com pare the ‘iterated-colim it’ construction o f the associated sheaf functor [H eller and Rowe 1962, Johnstone 1974]. Indeed, this works, up to a point: Proposition G iven a frame A , define an ordinal sequence o f frames N J A ) and h om o­ m orphism s cl : N ^ A ) - * N X(A)

by

N 0(v4) = v4, cS = id ^, N a +l (A) = N W M ) \ c j + 1 = c - c f ,

and J V ^ ) = linK N „(/l)

N M P < « < A)

51

Sublocales and sites

if A is a lim it ordinal, the c \ being the canonical m aps from the vertices o f the indicated diagram to its colim it. If there exists an ordinal a such that c“+ x is an isomorphism, then N X(A) is the free com plete Boolean algebra generated by A. P ro o f Proposition 2.9 show s that N a(A) must be Boolean. Corollary 2.9 plus induction (and the defining property o f colim its) show that every frame h om o­ m orphism from A to a com plete B oolean algebra factors uniquely through c°.

□

H owever, the left adjoint to the inclusion CBool-^Frm cannot exist, since Frm is an algebraic category and CBool is not (Theorem 1.2 and Proposition I 4.10): if A is the free frame on X 0 generators, its reflection in CBool would also be free on X 0 generators, which is im possible. So we conclude Corollary There exists a frame A for which none o f the m aps c j +1 : N a( A) ->Na+l (A) is an isom orphism .

□

N o w it follow s from P roposition 2.7(ii)plus induction that the m aps c® : A - * N a(A) are all epim orphism s in Frm; i.e. the N a(A) are all subobjects o f A in Loc (though not, o f course, sublocales). W e thus conclude that the locale A o f the Corollary has a proper class o f non-isom orphic subobjects; in the usual categorical term inology, the category Loc is not well-powered.

2.11 We now introduce an im portant generalization of Theorem 1.2, which will enable us to give explicit descriptions of frames specified by ‘generators and relations’. As in 1.2, we shall assum e th at our generators form a meet-semilattice; to handle the relations we wish to consider, we introduce a new concept. Let A be a meet-semilattice. By a coverage on A we mean a function C assigning to each a e A a set C(a ) of subsets of j(d), called coverings of a, with the following ‘meet-stability’ property: S e C(a) ^ { s a b\s e S} € C(b)

for all b ^ a .

F o r example, if A is a distributive lattice, we could take C{a) to be the set of all finite sets with join a ; meet-stability in this case is just the distribu­ tive law. By a site we m ean a meet-semilattice equipped with a coverage. We say a frame B is freely generated by a site (A , C) if there is a meetsemilattice hom om orphism f : A - + B which ‘transform s covers to joins’ in the sense that for every a e A and every S e C(a) we have f ( a ) = V { f ( s ) \ s eS} , B

and which is universal am ong such maps, i.e. every f ' : A - + B ' satisfying

58

I I : Introduction to locales

the same conditions factors uniquely through / by a frame hom om or­ phism G iven a coverage C on a semilattice A, we define a subset I of A to be a C-ideal if it is a lower set and satisfies (3 S e C { a ) ) ( S ^ I ) ^ a e I for all a 6 A. We write C-Idl (A) for the set of all C-ideals of A , ordered by inclusion. Proposition

F or any site (A , C), C-Idl (A) is a frame, and is freely generated by (A, C).

P roo f First we show that C-Idl (A) is a sublocale of the free frame D A constructed in 1.2. It is clear that an arbitrary intersection of C-ideals is a C-ideal, so if we define j : D A - + D A by j(S) = f ) { I e C - l d \ ( A) \ l = >S }

then we have S^ j ( S ) =j ( J ( S) ) for any S, and the image of j is precisely C-Idl (A). So we need only show that j preserves finite intersections. Let S, T e DA and write I for j ( S n T). Consider J = {a € A|(V s e S)(a

a s

el)} \

it is clear that since S n T ^ I . We shall show th at J is a C-ideal. Suppose U € C(a), U then for every s e S we have {u as|w e U } e C( u a s ) by meet-stability of C, and { u A s \ u e U } ^ I by the definition of J. Since / is a C-ideal, we deduce a a s e I for all s e S, and hence a e J. N ow if we define K = { a e A \ ( S t e J ) ( a A t e I )},

then a similar argum ent shows K is a C-ideal, and S But now we have

since S n J e / .

j(S)nj(T)=KnJ=I=j(SnTy9

the reverse inclusion is trivial since j is order-preserving. So j is a nucleus, and C-Idl (A) is a sublocale of DA. Now it is clear that for any S e C(a) we have a ej {[j {i(S)\s ^ S}),

so that the composite m ap A

D

(DA) j = C-Idl (A)

Sublocales and sites

59

transform s covers to joins. But if / : A -+ B is any other semilattice hom o­ m orphism from A to a frame with this property, it is easy to verify that the right adjoint g : B-+DA of the unique extension / : D A-+ B of / to a frame hom om orphism (1.2) is given by g( b) ={a e A\ f ( a ) Hb ] ,

and since / transform s covers to joins this set is a C-ideal, i.e. g factors through C-Idl(A). So / factors (uniquely) through j : DA-+C-ld\{A). □ As a particular case of the Proposition, we note Corollary

The set Idl {A) of ideals of a distributive lattice A forms a frame under the inclusion ordering. M oreover, the assignment A h- Idl (A) defines a left adjoint to the forgetful functor Frm-»DLat. P roo f Take C to be the coverage on A defined by finite joins, as described earlier. Then a C-ideal of A is ju st an ideal in the usual sense; and a meet-semilattice hom om orphism A -+ B transform s covers in C to joins iff it is a lattice hom om orphism . □

We shall study the frames which are freely generated by distributive lattices in the next section.

2.12 We conclude this section with another application of P ro p ­ osition 2.11: the construction of coproducts in Frm (equivalently, of products in Loc). Let ( Ay\y e T) be a family of frames, and write B for the set-theoretic product of the A r (Of course B is a frame, and is the product of the A y in Frm.) For each y, the projection p y \ B - + A y has a right adjoint qy : A y-+B, which sends a e A y to the unique element b with p y(b) = a and pd(b)— 1 for 6 y; and qy of course preserves meets. Let A be the sub-meetsemilattice of B generated by the union of the images of the qy, i.e. the set of all b e B such that py{b) = 1 for all but a finite num ber of indices y\ then it is easy to see that the m aps qy : A y-+A m ake A into the coproduct of the A y in SLat. (If we think of the A y as being the open-set lattices of topological spaces X y, then we can identify the elements of A with 'open rectangles’ in the product space Y \ X y ~ i*e*the sets which form a base for the Tychonoff topology.) From the universal property of coproducts, it is clear th at we should

60

I I : Introduction to locales

have a meet-semilattice hom om orphism from A to the coproduct of the A y in Frm, which is universal am ong hom om orphism s / such th at each of the composites f - q y preserves joins. We accordingly define a coverage C on A, as follows: if a e A and S £ A y, define S[y, a ] to be the set of all elements of A obtained on replacing the yth entry of a by a mem ber of S. Then define C(a)= {S[y, d]\y e T, S G A y and V S = py(a)}.

It is easily verified that C satisfies the meet-stability condition of 2.11, and th at a meet-semilattice hom om orphism / from A to a frame transform s covers in C to joins iff each of the composites / • qy preserves joins. So we have Proposition The coproduct of ( Ay\y e T) in Frm is C-Idl (A), where C is the

coverage defined above on the meet-semilattice coproduct A of the A r

□

W hen we think of the A y as locales rather than frames, we shall write H i ( Ay\y e T) for their product in Loc (i.e. their coproduct in Frm), to distinguish it from their product in Frm. Similarly, we shall write A x x tA 2 for a product of two locales. 2.13 Let (A^l yeT) be a family o f sober spaces. H ow does the locale product of the open-set locales Q(Xy) com pare with the openset locale of the product space As we observed in 2.12, we may identify elements of the semilattice coproduct of the Q { X y) with open rectangles in the product space; this gives us a m ap from the semilattice coproduct to Q(]~[Xy), which is easily seen to preserve finite meets and transform covers in C to joins. So we get a frame hom om orphism (j) : [ I , ( 0 ( X y)\y e r )-> Q (n (^y |y e H).

M oreover, (f) is surjective, since any open set in Y [ X y can be w ritten as the union of the open rectangles which it contains, and the latter clearly form a C-ideal. From the adjunction Q H p t of 1.4, we know th at the functor pt preserves products; and since the locales Q ( X y) are spatial (1.5), it follows easily th a t Y \ X y is (hom eom orphic to) the space of points of Y \ t (Q(Xy)). Thus we have

Sublocales and sites

61

Lemma Let (X y|y e T) be a family of spaces. Then Y[i (fi(Xy)) is isom orphic

to Q ( f |X y) (i.e. the m ap (j) above is an isomorphism) iff it is a spatial locale. □ There is one im portant case in which the conditions of the Lem m a are always satisfied: Proposition Let X and Y be spaces, and suppose X is locally com pact. Then Q(X) x ,Q(Y) is isom orphic to £l(X x 7). P ro o f We have to show that the com parison m ap (f> defined above is one-to-one; equivalently, to show for any open U ^ X x Y that if R is any C-ideal of open rectangles whose union is 17, then every open rectangle contained in U is a member of R (so th at R is determ ined by U). Let L7j x U 2 be an open rectangle contained in U. F o r each x e U u we can find a com pact (not necessarily open) neighbourhood K x of x which is contained in U 1. Then for each (x', y) e K x x U 2, we can find a rec­ tangular open neighbourhood Vx^y x W x- y of (x \ y) in U x x U 2 which is a member of R , since R covers U. F or each fixed y, the sets {J^ -Jx ' e K xj cover K x, and so we can find a finite subcover by sets Vx >y, . . . , VXnty. Let W y=CYi =i W x . y; then since R is a lower set we have Vx . y x W y e R for each /, and since R is a C-ideal we deduce (int K x) x W y e R . But the sets {Wy\y e U 2} cover U 2, and so by another application of the C-ideal axiom we deduce (int K x) x U 2 e R . But this is true for any x, and the sets {int K ^ x e U i } cover U u so a third application of the axiom yields U1x U 2e R . □

f2.14 W ithout som e such assum ption as local com pactness, the result of Proposition 2.13 is false: Proposition Let Q be the set o f rational numbers, topologized as a subspace o f R. Then the locale Q(Q) x ,Q(Q) is not spatial. Proof For notational convenience, we shall work not with Q but with the set 0> o f dyadic rationals (i.e. rationals o f the form m/2"), which is o f course hom eom orphic to Q. For each point (x, y ) = ( a / 2 m, b/2n) o f D x D, let Sx y be the open square o f side l/2 m+n+1

centred

on

(x, y). Thus S 0i0 = ( —1/4, 1/4) x ( —1/4, 1/4), S 1/2. 1/4 =

62

I I : Introduction to locales

(15/32, 17/32) x (7/32, 9/32), etc. Let R be the C-ideal o f open rectangles generated by { Sx y|(x, y) e O x 0 } ; then it is clear that (J K = D x D , since every point o f O x O is in som e Sx^y. We shall show that R is not the m axim al elem ent o f 0 ( 0 ) x ,0 ( 0 ). T o do this, we first consider how R m ay be constructed. Let R 0 be the downward closure o f {Sx>>.|(x, y ) e O x O} in the poset A o f open rectangles, and define an ordinal sequence (Ra) o f subsets o f A by induction: R a +1= {a g A\{3S g C ( a ) K S £ K a)} if ?. is a lim it ordinal. Then ( R%) is an increasing sequence o f subsets o f A, and m ust be eventually constant since P A is a set. But if R a, = R a + l , then C-ideal containing

is a C-ideal, and it is clearly the smallest

y|(x, y j G O x O}.

Suppose R contains all open rectangles. Then there is a least a (say a0) for which som e rectangle o f the form ( — 1/3, 1/3) x ( — 0 is in R a; and we can choose a S0 such that ( - 1 /3, 1/3) x ( - DLat).

4.6 W e may recover the original topology on a coherent space X from the patch topology together with the specialization ordering on its points. Lemma Let x and y be points of a coherent space X with x £ y. Then there is a com pact open subset of X containing x but not y. P ro o f Regarding x and y as prime filters of the lattice A of com ­ pact open subsets of X , let a be any element of x —y. Then the com pact open subset determ ined by a (i.e. by the principal ideal 1(a)) has the required property. □ Proposition Let (X , Q) be a coherent space, with patch topology Q'. A subset U of X is Q-open iff it is Q'-open and an upper set in the specialization ordering on X . P roof If U is Q-open, then it is certainly Q '-open and an upper set. Conversely, suppose U is Q '-open and upper, and let x e U. F o r each y e X — U we have x ^ y , and so there exists a com pact Q-open Vy con­ taining x but not y. The sets { X — Vy\y e X — U} are thus Q'-open and cover X — U, so by compactness of (X , Q ) there exists a finite subcover { X —Vyi, . . . , X — Vy} . Then f ] ni=, Vy, is an Q-open neighbourhood of x contained in U. □

4.7 Now suppose we are given a Stone space (X, Q') and a partial ordering on the points of X . Do the Q'-open upper sets define a coherent topology on X ? N ot necessarily: if we choose a sufficiently lunatic partial ordering, the topology need not be sober or even T0. Example

Let a N denote the one-point com pactification Nu{c o } of the discrete space fol= {0, 1, 2, . . . } of natural numbers, a N is a Stone space, as may easily be verified. If we give it the obvious total ordering 0 < 1 < 2 < ■• • < g o , then the process described above does yield a coherent space;

74

I I : Introduction to locales

b u t if we modify the ordering by interchanging 0 and g o , we get a nonsober space, (a N —{0} is closed and irreducible, b u t not the closure of any point.) The example shows th a t we cannot hope to characterize the ‘good’ partial orderings on Stone spaces by purely order-theoretic properties (for example, the property of having directed joins); it is essential to consider the way in which the order is related to the topology. We shall say that an ordered topological space is totally order-separated if, given any two points x, y with x ^ y , there is a clopen upper set con­ taining x but not y. (Clearly, a totally order-separated space is totally separated.) By an ordered Stone space we shall m ean a com pact space equipped with a partial order which makes it totally order-separated. Proposition Let (X, Q', ^ ) be an ordered Stone space. Then the set Q of Q '-open upper subsets of X is a coherent topology on X , and ^ is the specialization ordering for this topology. P ro o f First we show that the Q '-clopen upper sets form a base for the topology Q. Let U be Q-open, x e U; then for each y s X — U we have x ^ y, and so we can find a clopen upper set Vy containing x b u t not y. The argum ent of Proposition 4.6 now enables us to find a finite cover of X — U by sets X — Vn , . . . , X —Vyn, and hence a clopen upper set V= P)"= l Vyi such that x e V ^ U . N ow since the Q'-clopen upper sets are Q'-com pact, they are also com pact for the coarser topology Q; and every Q -com pact Q-open set m ust be of this form since it can be covered by finitely m any such sets. Thus it is clear th at K ( Q) is closed under finite intersections, and forms a base for the topology Q. N ow for any x e X, it is clear from total order-separatedness th at J,(x) is Q-closed, and since all Q-closed sets are lower sets it m ust be the Q-closure of {x}. Thus ^ is the specialization ordering for Q; so it only remains to show that (X , Q) is sober. The T0 axiom is again immediate from total order-separatedness; so suppose we have an Q-closed set C which is not the closure of any of its points. Then for each x e C we can find y e C with y ^ x , and hence a lower Q'-clopen Vx with x e Vx, y $ Vx. The sets {Vx\x e C} form an Q '-open cover of C, so by compactness there is a finite subcover {VXl, . . . 9 VXn}. But now the sets Vx . n C are proper closed subsets of C whose union is C; so C is reducible. □

Stone spaces

75

4.8 *Theorem The category CohSp of coherent spaces and coherent maps is isom orphic to the category OStone of ordered Stone spaces and orderpreserving continuous maps.

P roo f The last three propositions have proved that part of the theorem which pertains to objects of the two categories. So it only remains to show that a m ap f : X - * Y between coherent spaces is (continuous and)

coherent iff it is continuous for the patch topologies and order-preserving. But if / is coherent, then f ~ x preserves the basic open sets of the patch topology, and (like any continuous map) / is order-preserving. C on­ versely if / is patch-continuous and order-preserving, then / " 1 preserves upper patch-clopen sets; but we saw that these are exactly the com pact open sets of the coherent topology. □

4.9 To conclude, we present an interesting corollary of Theorem 4.8. It has two forms, one topological and one lattice-theoretic. *Corollary (i) A coherent space is a Stone space iff it is Tt ; thus we may substitute 'T[ for ‘H ausdorff in condition (iv) of Theorem 4.2. (ii) A distributive lattice is a Boolean algebra iff all its prime ideals are maximal. (Com pare I 2.6.)

P r o o f (i) If a coherent space (X, Q) is Tu then the ordering on its patch space (X, Q') is discrete; thus Q = Q'. (ii) The closed points of spec A are ju st the maximal ideals of A ; so this follows from (i). □

Exercise Let A be a Heyting algebra. Show th at the assignments F F c \ A G k { a g /4|H ~1 a g G}

define a bijection between maximal filters F of A and maximal ( = prime) filters G of *Hence obtain an alternative proof of part (ii) of the C orollary for Heyting algebras.

76

I I : Introduction to locales

Notes on chapter II Section 1 As indicated in the Introduction, the study of topological prop­ erties from a lattice-theoretic viewpoint was initiated by W allm an [1938] and pursued by M cKinsey and Tarski [1944, 1946], N obeling [1948, 1954] and Lesieur [1954]. It was C. Ehresm ann [1957] and his student J. B enabou [1958] who first took the decisive step of regarding com plete Heyting algebras as ‘generalized to p o ­ logical spaces1 in their own right. They called such lattices lo c a l lattices’; the term 'fram e’ was introduced by C. H. Dowker, who has studied them in a long series of jo in t papers with D. P apert Strauss [D ow ker and P apert 1966, 1967; Dowker and Strauss 1972, 1975, 1976, 1977], J. R. Isbell, in a path-breaking paper [1972], pointed out the need to introduce a separate term inology for the opposite of the category of frames, an d coined the term io cale’. Isbell was also the first to point out th at w hereas ‘previous work . . . seems to suppose th at topology is em bedded unchanged in the enlarged system . . . [in fact] one can adum brate num erous ch an g es. . . [and] at least one is a considerable im p ro v em en t. . . ’; thus he was the progenitor of the idea, which forms a large part of the raison d'etre of this book, th at the category of locales is actually m ore convenient in m any ways than the category of spaces. Free frames (1.2) were first constructed by Benabou [1958]; see also [Isbell 1972], T he adjunction of Theorem 1.4 was developed by P apert and P apert [1958], and again described m ore succinctly by Isbell [1972]. Isbell used the term ‘prim al locale’ (inspired by Exercise 1.5) for w hat we now call a spatial locale; he also used the term ‘prim al space’ for w hat we call a sober space. (The latter term is due to G rothendieck; see [G rothendieck and D ieudonne 1960], 0 2.1.1, and [G rothendieck an d Verdier 1972], IV 4.2.1.) T he first published characterization of spatial locales (not, of course, under th at name) seems to have been given by Buchi [ 1952]; see also [S. P apert 1959a]. T he notion of T0-space (Exercise 1.7) was introduced by Aull and T h ro n [1963]; it provides a useful generalization of the T { axiom in m any contexts where one is prim arily interested in sober spaces. T he partial order induced by a T0 topology has been studied by m any authors over the years; the term ‘specialization’, which we use for it in paragraph 1.8, comes from algebraic geometry (see [G rothendieck and Verdier 1972], IV 4.2.2). Alexandrov [1937] studied spaces whose topology was m axim al com patible w ith a given ordering (he called them ‘discrete spaces’), and proved the result of Exercise 1.8. D obbs [ 1980] has investigated posets for which the Alexandrov and upper interval topologies coincide. T he Scott topology (1.9) derives its nam e from [Scott 1972], where it was extensively used for the first tim e; but a particular case of it appears earlier in [D ay an d Kelly 1970]. The counterexam ple in Exercise 1.9 is due to Johnstone [1981a] (see also [Isbell 1982]); P roposition 1.10 is due to Scott [1972]. The partial order on th e hom -sets of Loc (1.11) was studied by Isbell [1975a].

Section 2 Sublocales (quotient frames) have been studied by several authors, notably D ow ker and P apert [1966] and Isbell [1972], T he term ‘sublocale’ is due to

N o te s on chapter II

11

Isbell, who also used 'p a rt’ to m ean approxim ately the sam e thing. The use of nuclei as a tool for studying sublocales (and the term ‘nucleus’ itself) was initiated by H. Sim m ons [1978] and his student D. M acnab [1981]; they were, however, strongly influenced by the work of Lawvere and Tierney [Law vere 1972] in reducing the n otion of a G rothendieck topology to th at of an ‘internal’ nucleus on the subobject classifier in a topos. O pen and closed nuclei (2.4) com e from the sam e source [Tierney 1973], via w ork of G. C. W raith [1975]; Lem m a 2.4 is due to Isbell [1972], who used the term ‘limitless’ for locales which are Boolean algebras. P roposition 2.5 is due to Isbell [1972], though his proof was different from ours; it involved a rather com plicated argum ent (based on P roposition 2.7) to verify the infinite distributive law in N(A). O u r explicit construction of the im plication in N ( A ) appears to be new. The results of paragraphs 2.6-2.10 are all in [Isbell 1972]. T here are several alternative constructions of the em bedding required for C orollary 2.6; for two different ones, see [F unayam a 1959] and [Jo h n sto n e 1977], Rem ark 7.56 (the latter being based on an idea of Freyd [1972]). Beazer and M acnab [1979] and Sim m ons [1980, 1980a, 1982] have investigated frames A for which the ordinal sequence ( N a(A)) of P roposition 2.10 converges after one or two steps. T he introduction of sites (2.11) as a system atic m eans of constructing locales appears to be due to the au th o r [Johnstone 1979, 1981c], although m any special cases of th e construction (for example, th a t in C orollary 2.11) were well-known previously. T he term ‘site’ is of course borrow ed from G rothendieck and Verdier [1972], II 1.1.5; though in adapting G rothendieck’s notion o f ‘(pre)topology’ on a category to o u r notion o f ‘coverage’ on a semilattice, we have deliberately om itted th e ‘com position-stability’ axiom (PT 2 of [G rothendieck and V erdier 1972], II 1.3). This change was originally m otivated by consideration of examples like that in 2.12, b u t it proved useful for other reasons when we came to consider internal sites in a to p o s [Johnstone 1979]. The term ‘coverage’ was suggested to the author by S. M ac Lane. P roduct locales (coproduct frames) have been studied by several authors. O u r construction (2.12) is very sim ilar to th at given by D ow ker and Strauss [1977], ap art from the correction of a m inor error in the D ow ker-S trauss paper and the use of sites to m otivate the construction. Am ong other constructions, let us m ention the interesting observation by D. W igner [1979] th at A x m ay be viewed as the lattice of G alois connections between A and B. P roposition 2.13 is a folk-theorem ; it m ay be found in [D ow ker and Strauss 1977] (for H ausdorff spaces) and [Isbell 1981] (together with a partial converse), and a close relative of it appears in [Jo h n sto n e 1977] as Exercise 7.4. (However, it was evidently not known to M. Hakim [1972], since she quoted a m uch weaker result as Rem ark II 3.2.) The result of P roposition 2.14 was know n to Isbell [1972], though our explicit construction of a nontrivial C-ideal containing all the points of 0> x 0> appears to be new; Isbell used the argum ent which we shall sketch in III 1.4. M ore com plicated counter­ examples to th e isom orphism between f |,( Q ( X y)) and Q (fl [D ow ker and Strauss 1977].

m ay ^ f° und in

78

11'* Introduction to locales

Section 3 T he study of finite elem ents in com plete lattices was initiated by Birkhoff and Frink [1948], who introduced the term ‘inaccessible’ for elem ents satisfying condition (iv) of 3.1. Subsequently, it was realized th at conditions (i)—(iii) were m ore fundam ental; Diener [1956] introduced the rather clumsy term ‘intranscessible’ for elem ents satisfying these conditions. T he term ‘com pact elem ent’ has also been used, for obvious reasons; we prefer to call such elem ents simply ‘finite’, since there does not seem to be any danger th at this term will lead to con­ fusion. T he term ‘coherent’, which we use for locales (3.2) and spaces (3.4), has a rather varied history; our use of it m ay be traced back to the ‘coherent sheaves’ of Serre [1955] via the ‘coherent toposes’ of G rothendieck and Verdier [1972]. There is also a connection with the ‘coherent logic’ of Joyal and Reyes [Reyes 1974, M akkai and Reyes 1977], which we shall meet in section V 1; coherent locales are exactly the ‘L indenbaum algebras’ of coherent propositional theories. (In this context, Theorem 3.4 may be viewed as the G o d el-H enkin com pleteness theorem [G odel 1930, H enkin 1949] for such theories; cf. [H enkin 1954].) C oherent spaces are know n by several o ther nam es: ‘spectral spaces’ [H ochster 1969], ‘quasi-Boolean spaces’ [H ofm ann an d Keimel 1972], and even ‘Stone spaces’ [Balbes and Dwinger 1974]. C orollary 3.4 m ade its first appearance in [S tone 1937b]. W e deduced it from Theorem 3.4, which m akes use of the P IT for distributive lattices; on the other hand, if we interpret the C orollary as asserting not merely th at a duality exists b u t th at the specified functors give rise to it, then it implies the P IT (since any nontrivial distributive lattice m ust have a nonem pty spectrum). In contrast, Lem m a 3.5 requires the full strength of the M IT for distributive lattices. Lem m a 3.6 is due to W allm an [1938]; note th at m any authors (e.g. C ornish [1972]) use ‘norm al' to m ean the dual of th e property we consider (i.e. they consider norm ality as a property of the lattice of closed sets of a space, rather than the lattice of open sets). P roposition 3.7 m ay be found in [Sim m ons 1980b]; cf. also [D e M arco and O rsatti 1971]. (Observe th a t the im plications (i)=>(ii) and (i)=>(iii) of P roposition 3.7 do not require the axiom of choice; thus the M IT for norm al distributive lattices is no stronger than the PIT.) Like max B, the subspace m in B of spec B whose points are the m inim al prim e ideals (or m axim al filters) of B has been studied by a num ber of authors, mostly in th e ring-theoretic context which we shall m eet in section V 3; see for example [H enriksen and Jerison 1962, 1965a], [Speed 1969] and [H ochster 1971]. (The topological properties of m in B are rather different from those of m ax B ; it is always H ausdorff (indeed, zero-dim ensional), b u t not necessarily com pact.) There is a similar, though n ot identical, result to P roposition 3.7 for m inim al spectra; see [K ist 1974] and [Sim m ons 1980b].

Section 4 T otal disconnectedness and related properties were first studied by Sierpiriski [1921] and K naster and K uratow ski [1921]. In texts on topology, there is considerable confusion about the nam es of the three properties introduced in 4.1; doubtless this is because they are equivalent in com pact H ausdorff spaces.

N o te s on chapter II

79

O u r term inology follows that of Steen and Seebach [1970]. O u r proof of the im pli­ cation (i)=>(ii) in T heorem 4.2 is taken from [H urew icz and W allm an 1941]; it is interesting to note that m any books on topology prove this fact (or the equivalent statem ent I n a com pact H ausdorff space, every quasicom ponent is a com ponent’) by a different argum ent involving Z o rn ’s Lem m a. (Indeed, H ocking and Y oung [1961] go so far as to say (p. 45) T h e com plete reliance upon the m axim al principle in these proofs is apparently unavoidable’ - though, to be fair, it is not clear w hether this rem ark applies to the proof of the result quoted above, or only to the two proofs which follow it.) T he counterexam ple in Exercise 4.3(i) is a famous one, due to K naster and K uratow ski [1921], F or the history of C orollary 4.4, see the Introduction (and note th at it, like C orollary 3.4, implies the PIT). P roposition 4.5 is due to H ochster [1969], who proved it by direct topological m ethods. In relation to Exercise 4.5, it should be m entioned th at Perem ans [1957] has show n w ithout the axiom of choice th a t any distributive lattice is isom orphic to a sublattice of the free Boolean algebra which it generates. Priestley [1970, 1972] introduced the notion of ordered Stone space (4.7) and used it to obtain a modified version of Stone duality for distributive lattices; she thus implicitly proved Theorem 4.8. (Priestley used the term ‘totally order-disconnected’ where we have ‘totally order-separated’; the latter is m ore in keeping with our term inology adopted in paragraph 4.1.) The proof of Theorem 4.8 was m ade m ore explicit by C ornish [1975]; independently, Joyal [1971, 1971a] had also determ ined the relation between coherent spaces and ordered Stone spaces. (See also [S tralka 1980].) C orollary 4.9(ii) is due to N achbin [1947, 1949a], who gave a lattice-theoretic proof of it. N o te that it is equivalent to the P IT ; for it implies that every non-B oolean distributive lattice has a prim e ideal, and every nontrivial distributive lattice may be em bedded in a non-Boolean one by the m ethod of Exercise I 4.8. Finally, let us rem ark th at Exercise 4.9 shows that the M IT for the duals o f Heyting algebras is no stronger than the P IT (although, as we rem arked earlier, M IT for Heyting algebras implies the axiom of choice).

I ll C om pact H ausdorff spaces

1. Compact regular locales 1.1 Although our m ain goal in this chapter is to prove two im por­ tan t theorem s (those of M anes and Gleason) ab o u t com pact H ausdorff spaces, our general philosophy leads us to begin it by taking a look at com pact H ausdorff locales. N ow there is no difficulty in saying w hat it m eans for a locale A to be com pact ; we simply say th at its top element 1A is finite in the sense of II 3.1. But it is less easy to state the H ausdorff property w ithout m entioning points; although it can be done (see [Sim m ons 1978a]), the resulting axiom seems very inconvenient to work with. W e therefore sidestep the problem by using the well-known fact that a com pact H ausdorff space is regular, and conversely a regular T0-space (in particular a regular sober space) is Hausdorff. Let a, b be elements of a distributive lattice A. We say a is well inside b (and write a ^ b ) if there exists c e A with c a £j = 0, c v b = 1. (If A is a H eyting algebra - and in practice it will invariably be a locale - this relation holds iff ( ~ \ a ) v b = 1, since ~\a is the largest element disjoint from a \ but the form which we gave as the definition shows at once that the relation ^ is preserved by lattice hom om orphism s (e.g. frame hom o­ m orphism s) even if they do not preserve “ I. However, we note in p ar­ ticular that in the open-set lattice Q(X) of a space X, the relation U < V holds iff the closure of U is contained in V, since "1 U is the interior of the com plem ent of U.) First we note some trivial properties of the relation < :

Lemma (i) a ^.a iff a has a complement.

(ii) a ^ b implies a ^ b . (iii) a ^ b ^ c ^ d implies a ^ .d . (iv) F o r any a e / t , { 5 e A\a ^ b } is a filter o f A , and {c e A \c < a ) is an ideal. 80

Compact regular locales

81

P roof (i) is trivial. F or (ii), suppose a A c = 0 and b v c — l; then a = a a [b v c ) = (a a b) v ( a Ac) = a a b, so a ^ b . (iii) is again trivial, and (together with (i)) it at once implies that {b\a < b} is an upper set and contains 1. Suppose a l^ .b 1 and a < b 2\ let c u c2 be such th at a ACj = 0 and bi VCi — l. Then we have a a (ci v c 2) = ( a A c 1) v ( f l A c 2) = 0 v 0 = 0

and (bi a b 2) v ( c i v c 2) = ( b t v c , v c 2) A ( b 2 v c t v c 2) = 1 a 1 = 1, so a < (bi a b 2). Thus {b\a^.b} is a filter; the proof th at { c \ c ^ a } is an ideal is similar. □ W e say that a locale A is regular if every a t A satisfies a — V {b ^ A \b ^.a}, A

i.e. every element of A is the join of the elements well inside itself. F o r the locale Q(X), this says that every open set U can be covered by open subsets whose closures are contained in U, which is easily seen to be equivalent to the usual definition of regularity for the space X . Exercise Define zero-dimensionality for locales (cf. II 4. l(iii)), and prove th at a zero-dimensional locale is regular. Hence show (without m entioning points) that the locale of ideals of a B oolean algebra is regular.

1.2 topological results.

Next we prove the localic analogues of some well-known

Proposition (i) A sublocale of a regular locale is regular. (ii) A closed sublocale of a com pact locale is compact. (iii) A com pact sublocale of a regular locale is closed. P roo f (i) Let j be a nucleus on a regular locale A, b e A y If a ^.b in A , then there exists c with a A c = 0, b v c = 1, and so j(a) A j( c ) = j( 0 \ b v j ( c ) = 1, which implies that j( a ) ^ .b in A y Now b ~ V {a e A \ a < Ab} A

s: V {i(a)|a ^ Ab] A

^ V {a' t A j \ d <Ap}> aj

so A j is regular.

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I I I : Com pact H ausdorff spaces

(ii) If 1A is finite in A , then it is clearly also finite in A c(a)— T(a)(iii) Let A j be a com pact sublocale of a regular locale A. By cutting down if necessary to a closed sublocale of our original A, we may suppose A j is dense in A. Let a be an elem ent of A with j(a )== 1; then we have a = V A {b e A \ b ^ a } , and so U = v {j(b)\b S et reflects isom or­ phisms only when restricted to the subcategory of spatial locales. Accord­ ingly, our approach to M anes’ Theorem will be m ore conventionally ‘topological’ than the line we have followed so far. F irst we need to recall some facts about convergence of filters on topological spaces. Let X be a set; as is custom ary, we say ‘filter on X 5 (note: on , not in) for ‘proper filter in P X \ and ‘ultrafilter on X ’ for ‘maximal filter in P X \ F or any x e l , the principal filter *fx = |({x}) is an ultrafilter on X . If we are given a topology on X , we define the neighbourhood filter N(x) to consist of all U e l such that x is in the interior of U. We say x is a limit of a filter F if F 2 iV(x). *Proposition

(i) A topological space X is H ausdorff iff every ultrafilter on X has at m ost one limit. (ii) A space X is com pact iff every ultrafilter on X has at least one limit. P r o o f (i) Suppose X is Hausdorff. Then if x ^ y 9 N(x) and N(y)

contain disjoint sets, and so cannot bo th be contained in the same ultra­ filter. Conversely, suppose x and y have no disjoint neighbourhoods. Then the set { U n V\ U e N ( x \ V e N(y)} is a filter on X , and we may enlarge it to an ultrafilter by Lem m a I 2.3. But this ultrafilter m ust have both x and y as limits. (ii) Suppose X is compact. Then for any ultrafilter F on X , the inter­ section of all the closed sets in F is nonem pty, since every finite sub­ intersection is. Let x be a point in the intersection. Then since x is in the closure of every mem ber of F, every neighbourhood of x meets every m em ber of F ; so { U n V \ U e F, V s N ( x ) } is a (proper) filter on X . But since F is maximal, this filter m ust be F itself; hence N( x) ^ F . Conversely, let { U a|a e A} be an open cover of X with no finite subcover. Then the sets ( X — U a\ of e A, generate a proper filter on X , and we can extend it to an ultrafilter F by Lemma I 2.3. But F cannot contain any U aj and so cannot have any limit point in \J {C/a|oc e A} —X. □ 2.3 Proposition 2.2 shows that, if (X , Q) is any com pact H ausdorff space, then the operation of taking limits of ultrafilters defines a function

M anes’ Theorem

95

^ : p X - + X . M oreover, the proof of Proposition 2.2 shows that, for any ultrafilter F, we have P) {cl U\U e F} = {£(F)}. Since (X , Q) is regular, it follows easily that £(F) lies in an open set U iff there is an open set V s F with cl V ^ U ; th at is, r 1( U ) = { F e p x \ ( 3 V ^ U in Q )(F eF )}.

In particular, this shows that £ is continuous (since the sets { F \ V e F} are basic open in fiX); and on chasing through the proof of Theorem 2.1, we find th a t £ is actually the counit of the adjunction of Corollary 2.1. *Lemma Let X and Y be com pact H ausdorff spaces. A function f : X - + Y is continuous iff it preserves limits of ultrafilters, i.e. iff the diagram Pf f i x ------------- - ---------- >PY

{

f

commutes. P ro of N ote first that the m ap j3f is defined by Pf(F) = { V ^ Y \ f ~ l( V ) e F }

for F e PX. So if / is continuous, we have m m ) } = f ] { V ^ Y \ V closed, f ~ \ V ) e F }

and hence / - W ( F ) ) ) =>n { U s x | IU closed, U e F } = {£(F)}, i.e. /(^(F )) = ^(j?/(F)). Conversely, suppose / is not continuous; let V be an open set in Y and x a non-interior point of / _1(^)- Then the sets (U —f ~ 1(V)% U s N ( x ) , generate a proper filter on X , which we can enlarge to an ultrafilter F ; and x is clearly a lim it of F. But / ~ X(V) $ F, so V t Pf (F ) and hence /(x ) is not a limit of j?/(F). □

2.4 *Theorem (Manes) The category KHausSp of com pact H ausdorff spaces and con­ tinuous m aps is algebraic.

96

I I I : Com pact H au sdorff spaces

Proof Let B = (j5, rj, ju) be the m onad on Set induced by the adjunction of Corollary 2.1. The com parison functor K : KHausSp->SetB sends a space (X, Q) to the B-algebra (X , £), where £, : f i X ^ - X is the m ap defined in paragraph 2.3; so Lem m a 2.3 tells us th at K is full and faithful. To show that K is an isomorphism, we have to show th at every B-algebra structure on a set X is induced in this way by a (necessarily unique) com pact H ausdorff topology on X . Let (X , £) be a B-algebra. If U is a subset of X , we define

L/ = {£(F)|Fej?X , L /e F } ; we shall show that U \->U is a closure operation on X in the sense of K uratow ski [1922]. First note that for a principal ultrafilter rjx, we have U e rjx iff x e U; and since ^ is a B-algebra structure, we have ^(rjx)=x for all x. Hence U SL7. F o r any ultrafilter F, we have ( U u V) e F iff either £/ e F or F e F ; s o ( [ / u F ) = L / u K Now let F be an ultrafilter with U e F. We shall show th at there is an ultrafilter G with £(F) = £(G) and U e G, so th at £(F) e U and hence U = U . To do this, consider the com m utative diagram

where ju is the m ultiplication of the m onad B, i.e. the B-algebra structure induced by the Stone space topology on f$X. But the basic open sets in P X (corresponding to principal ideals in P X ) have the form S(t/) = {F e p x \ U e F} for some U e l ; and since these sets are actually clopen, we have Aii1(S ([/))= { ® e ^ X |S ([/)e « > } . Hence for any <J>e we have jux(<J>)= {U eX |S(L/) e <J>}. O n the other hand, we have {U e X |^ _ 1 (L/) e }. Now for each V e F, we have V n U = ^ 0 , and hence £,~1( V ) n S ( U ) ^ 0 . Hence the sets {£ ~ 1( V ) n S ( U ) \ V e F} generate a proper filter on which can be extended to an ultrafilter <J>; and by construction we clearly have = F and U e /ix(0). But by the diagram above we have = £(F), so jUxW *s the required ultrafilter on X . Thus {U £ : X \ U = U} is the lattice of closed sets for a topology Q on X. Let F be any ultrafilter on X , and suppose £(F) = x. Then x e V for every

M anes' Theorem

91

V e F, so (by the argum ent of Proposition 2.2(ii)) x is a limit point of F for the topology Q. Conversely, suppose y is a lim it point of F. Then for every V e F, we have y e V \ i.e. there exists an ultrafilter Gv with V e G v and £(Gv) = y. Thus the sets { S (F )n ^ _ 1 (v )|F eF } are all nonempty, and so they generate a proper filter on which can be extended to an ultra­ filter x¥. N ow we have ^ 1(y) e T and so j?Set. It is clear th at this functor creates arbitrary limits, since the forgetful functors K H ausSp->Set and A-*-Set both do so; so the only problem is to verify the solution-set condition. Let F : Set-* A denote the free functor for A, and consider a m ap f : X - + A from a set X to (the underlying set of) a com pact H ausdorff algebra A. Since A is an algebra, / extends uniquely to a hom om orphism / ' : FX-+A, whose image is the subalgebra A' of A generated by the image of f O f course A' need not be closed in A, b u t its closure A' is again a subalgebra, since for any cardinal n (finite or infinite) the closure of {A')n in the Tychonoff topology on A" is ju st (A ")n, and hence the n-ary operations of A map this set into A". Also, if we further extend / ' to a continuous m ap / " : p{FX)->A by the adjunction of Theorem 2.1, the image of f " is precisely A ". But A" is an object of

98

I I I : C om pact H a u sdorjf spaces

KHausA; thus we have show n th at any m ap / : X- >A as above factors through one for which the induced m ap p(FX)->A is surjective. But there is only a set of non­ isom orphic m aps X - + A w ith this property, since there is only a set of surjective images of p ( F X ) ,’ so the solution-set condition is verified. The rem ainder of the argum ent using Beck’s Theorem is straightforw ard. It is required to show that the forgetful functor KHausA->Set creates coequalizers for pairs of m aps which become contractible in Set. But if A=t B is such a pair, its co­ equalizer B->C in Set inherits a unique algebra structure and a unique com pact H ausdorff topology, since the forgetful functors A->Set and KHausSp-* Set are m onadic. So we need only show th a t the two structures are com patible, i.e. th at the algebra operations C "-*C are continuous. But this follows easily from the fact th at Bn->Cn is a quotient m ap of (com pact Hausdorff) spaces.

□

3. Gleason’s Theorem 3.1 In the last section, we studied the spaces fiX which are the free algebras in the algebraic category KHausSp. In other algebraic categories (particularly categories of m odules over a ring) we are accus­ tom ed to study a generalization of free objects, namely projective objects; it turns out that the projective objects in KHausSp are also of interest. F irst we recall the definition: an object P in a category C is said to be projective if, whenever we are given a diagram of the form P

f

with / an epim orphism in C, there exists h :P -> X with f h = g. M ore generally, we may consider E-projectives , for which the m orphism / in the diagram above is required to belong to a particular class of epim orphism s E (for example, the regular epimorphisms). Lemma

(i) Suppose C has pullbacks, and the class E is stable under pull­ back. Then an object P is E-projective in C iff every Em orphism / : Q - + P is a split epimorphism . (ii) F o r any E, a retract of an E-projective object is always Eprojective. P r o o f (i) The given condition is necessary, since we may take g

Gleason's Theorem

99

to be the identity m orphism on P in the definition of projectivity. To show it is sufficient, suppose given a diagram as above, and com plete the pull­ back square

Then / ' e E by assum ption; if h' : P ^ Q is a one-sided inverse for it, then the com posite g ' h is the required morphism . (ii) Suppose given a diagram P

+Q

■> Y

with / e E , P E-projective and ts = idQ. T hen we can find h : P - + X with f h = gt , and the com posite hs : Q ^ X has the required property. □ 3.2 Suppose we have a functor G :C -> D which has a left adjoint F. (We shall norm ally assume in addition th at G is faithful, and so reflects epimorphisms.) Then we may consider the class E G of m orphism s / in C such that G ( f ) is a split epim orphism in D. Proposition G iven F and G as above, an object of C is E G-projective iff it is a retract of an object in the image of F . Proof Suppose P is E G-projective. The triangular identity G(fi) *>?G= idG (I 3.4) implies th at the counit m ap eP : F G P -> P is in E G. So it m ust be a split epim orphism , and hence P is a retract of FGP. To prove the converse, it suffices by Lemma 3.1(ii) to show th a t objects in the image of F are EG-projective. But if we are given FT 9

100

I I I : C om pact H au sdorff spaces

with / e E g, we may transpose across the adjunction to get a diagram

T

Gf G X --------- -J------HjY

then if s : G Y - > G X is a one-sided inverse for G f the com posite // = s g : T - > G X satisfies G f ' h = g, and so its transpose h : F T - > X satisfies □

fli = 9-

We now specialize to the case when D = Set and C is an algebraic category. In this case the axiom of choice tells us th at every epim orphism ( = surjective function) in Set is split; and the m orphism s in C whose underlying functions are surjective are precisely the regular epim orphism s (P roposition I 3.8). So we obtain *Corollary In an algebraic category, an object is regular projective (i.e. p ro ­ jective with respect to the class of regular epim orphisms) iff it is a retract

of a free algebra.

□

3.3 In passing, we note Lemma Every epim orphism in KHausSp is surjective; thus ‘projective’ and ‘regular projective’ mean the same in this category. Pr oo f Let f : X - > Y be a m orphism in KHausSp, with image I £ Y, and let = j be the equivalence relation on Y defined by y = iy' & either y = y ’or (y € I and y' e I).

Since I is com pact and therefore closed in Y, it is easy to see th a t the quotient space Z = Y/ = 7 is (compact) Hausdorff, and so the quotient m ap q : Y-»Z is in KHausSp. Suppose X (and hence I) is nonem pty, and let p : Y-»Z be the constant m ap sending every element of Y to the equival­ ence class I. Then p f —q f and so if / is epi we m ust have p — q; i.e. I — Y and / is surjective. (This proof requires some slight modification if X is em pty; we leave this to the reader.) □

Gleason's Theorem

101

3.4 *Proposition

Every projective in KHausSp is a Stone space, and its Boolean algebra of clopen subsets is complete. P r o o f Let P be projective in KHausSp. By C orollary 3.2, we can

express it as a retract of a free object P X by m aps p : P - > [iX and q : say. It is then easy to verify that P - ^ P X = zpq $ p X

is an equalizer diagram in Sp; so P is a closed subspace of /?X, and hence a Stone space. It now follows at once from the Stone Representation Theorem th at the Boolean algebra B corresponding to P is a retract of P X ; so it suffices to prove th at a retract of a complete Boolean algebra is complete. But this is trivial, since if we have maps / : B -> A and g : A -> B in Pos with g f = idB and A is complete, then we may construct joins in B by setting V S=

{/(siseS}^.

□

3.5 To obtain a converse to Proposition 3.4, we m ust briefly digress to examine the topological properties of Stone spaces which correspond to complete Boolean algebras. We recall (Exercise 11.13) th a t the following four conditions on a H eyting algebra A are equivalent: (i) The identity " 1 (a A 6 )= _ la v " lfc holds for all a, b e A. (ii) The identity “ la v “ 1~~1a = 1 holds for all a € A. (iii) Every “ IH -stable elem ent of A has a complement, i.e. A - in coincides with the subset A c of com plem ented elements of A. (iv) A-\-\ is a sublattice of A. If A is complete, and in addition is zero-dim ensional as a locale (i.e. every element can be written as a jo in of com plem ented elements), then we may add a fifth equivalent condition to those above: (v) A c is a complete Boolean algebra. This is certainly implied by (iii), since A-\-\ is a sublocale of A and hence complete. O n the other hand, given (v), let a e A-\-\, and define b = V {a! € A c\a!^a}. Ac

Then we have b ^

{a! e A c\ a ' ^ a } = a . O n the other hand, if c is com ­

102

I I I : C om pact H au sdorff spaces

plem ented and b

a c

=

“ la, we have V fa'

A c \a '

e A c, a ' ^ a } = 0

Ac

since

a!

a b

c

^ a a "1 a = 0. So

a

~la = V

a

c|c € A c, c ^ a } = 0 ,

A

and hence “ I —Ia = a; i.e. a is complemented. We call a locale A extrem ally disconnected if it satisfies the equivalent conditions (i)—(iv) above; a space X is extremally disconnected if the locale Q( X) is (equivalently, if the closure of every open set in X is clopen). (The name ‘extremally disconnected’ is slightly inappropriate; for H aus­ dorff spaces, the condition implies at least total separatedness, b u t for non-H ausdorff spaces it doesn’t really have anything to do with dis­ connectedness.) From the rem arks above, we immediately deduce Lemma

F o r a Boolean algebra B, the locale Idl(B) (or *equivalently the space spec B) is extremally disconnected iff B is complete. □

3 .6 O ne more definition: we say th a t a surjection f : X - > Y in KHausSp is minimal if there is no proper closed subspace X ' ^ X such th a t the restriction of / to X ' is still surjective. *Lemma

If / : X-> Y is a surjective m ap in KHausSp, then there is a closed subspace X ' £ X such that f \ x . is a m inim al surjection. Proof Consider the set T of closed subspaces X ' ^ X such th at f \ X’ is surjective, ordered by inclusion. If S is a totally ordered subset of % then for each y € Y the sets {f-\y)nX'\X'eS}

are nonem pty closed subsets of the fibre / ” ^y), and (being totally ordered by inclusion) they have the finite intersection property. So by compactness ° f / - 1 (y) their intersection is nonem pty; th at is, the intersection of the members of S meets every fibre of f and hence is itself in T. So we can apply Z orn’s Lemma (downwards) to the ordered set T, to obtain a m inimal element X ’\ but then f \ x . is a minimal surjection as defined above. □

Gleason's Theorem

103

3.7 Lemma Let Y be com pact, H ausdorff and extremally disconnected. Then any minimal surjection / : X - > Y in KHausSp is a hom eom orphism . Proof Since any continuous m ap of com pact H ausdorff spaces is closed, we have only to show th at / is one-to-one. Suppose x u x 2 are

distinct points of X with f ( x l ) = f ( x 2) = y say. Since X is Hausdorff, we can find disjoint open neighbourhoods U u U 2 of x x and x 2. F o r U S X , define

V/([/)={y 6 y |/" 1(y)sC/}; since V /L /) is simply the com plem ent of the image of the com plement of U , the fact that / is a closed m ap implies th a t V / preserves open sets. So V /L ^ ) and V /L /2) are open in Y; and they are clearly disjoint, since U i and U 2 are disjoint and / is surjective. T h at is, we have ^ W J( U 1) n V j { U 2) ) = Y

in Q(Y); b u t since Y is extremally disconnected, we deduce th at n(V / ( [ / 1 ) ) u - |( V / ([ /2) ) = x and in particular th a t one of the sets y € N ow consider the set

"1

(V/tA)) m ust contain y - say

V = U , n f - 1n < S f ( U l)));

this is open in X and contains x u b u t V ^ F ) is clearly empty. So the restric­ tion of / to the proper closed subset X — V is still surjective, contradicting the minimality of / □ C om bining the last four results, we obtain *Theorem (Gleason) The projective objects in KHausSp are precisely the extremally disconnected spaces. Proof One way round is just Proposition 3.4 plus Lemma 3.5. Conversely, suppose Y is extremally disconnected; by Lemma 3.l(i) it suffices to show that every surjective / : X ^ > Y in KHausSp is a split epimorphism. Given such an f use Lem m a 3.6 to obtain a closed subspace X ' ^ X such that f \ x . is minimal. Then f \ x - is a hom eom orphism by Lem m a 3.7; com posing the inverse of this hom eom orphism with the inclusion X ' - * X yields a one-sided inverse for f □

104

I I I : Com pact H ausdorff spaces

t3 .8 Is it possible to extend G leason’s Theorem to a larger category of spaces th an KHausSp? O n exam ining the proofs of Lem m as 3.6 and 3 J , we see that they invoke ju st three properties of a m ap f \ X - > Y between com pact H ausdorff spaces: (i) F o r every y s Y , the fibre / “ 1(y) is com pact. (ii) / is a closed m ap, i.e. the function V/ : P X - > P Y defined in the proof of Lem m a 3.7 preserves open sets. (iii) D istinct points in the same fibre of / have disjoint open neighbour­ hoods in X - equivalently, the diagonal m ap A :A '- > A 'x yA 'is a closed embedding. We shall say th at a continuous m ap between arbitrary spaces is proper if it satisfies the above three conditions. *Exercise Show th at f \ X - > Y is proper if f y uniquely reflects convergence of ultrafilters’, i.e. for any ultrafilter F on X and any lim it y of fif(F) in Y, there is a unique x e / -1 (y) which is a lim it point of F. N ow if we replace the hypothesis th at X and Y are com pact H ausdorff in Lem m as 3.6 and 3.7 by the hypothesis th at / is a proper m ap, then both Lem m as rem ain true w ithout further alteration. (Of course, we also need to know th at the restriction of a proper m ap to a closed subspace of its dom ain is again proper, b u t this is a triviality.) To com plete the generalization of Theorem 3.7, we need to know th at we are in a position to apply Lem m a 3.1(i); th a t is, Lemma The class of proper m aps is stable under pullback in Sp. Proof Suppose we are given a pullback square

with / proper. First we note th at /i~ 1(y) is hom eom orphic to f ~ l (g(y)) for any y g Y, so condition (i) is immediate. F o r condition (ii), let U be open in P and suppose h~ 1(y)QC7. Then for any x t f ~ l (g(y)\ we can find open neighbourhoods Vx, W x of x and y respectively, such th at (Vx x W x) n P ^ U (here we are identifying P with the subspace {(x\ y' )\ f(x' )—g{y')} of X x 7), and since f ' ^ g i y ) ) is com pact we have a finite cover of it by sets Vxv . . . , VXn. Define

n n V=[j vx, w = [ \ w xii=1 i=l

Gleason's Theorem

105

then ( V x l V ) n P ^ U . Also, V is open and contains f ~ x(g(y% so V /K ) is open and contains g(y). Now g _1(V' j ( V ) ) n W is easily seen to be an open neighbourhood of y contained in Wh(U); so Wh{U) is open. Finally, for condition (iii) let p u p 2 be distinct points of P with the same image under h. Then k(p{) and k(p2) m ust be distinct points of X with the same image under / ; so we can find disjoint open neighbourhoods U u U 2 of them in X. Then k ~ l ( U i) and k~ i ( U 2) are the required open neighbour­ hoods of p l and p 2 in P. We can now generalize Theorem 3.7 as follows:

□

* Theorem Let S Prop denote the class of surjective proper m aps in Sp. Then the SProp-projectives in Sp are precisely the extrem ally disconnected spaces. Proof The proof th at extrem ally disconnected spaces are SProp-projective is now exactly as in Theorem 3.7. Conversely, suppose X is SProp-projective and let U be an open subset of X. Let W be the disjoint union (cl U)II (X —U), and let p : W- > X be the m ap obtained by com bining the tw o inclusion m aps (cl U) - * X and (X —U)-*X. It is easily verified th a t p is (surjective and) proper, so it m ust have a splitting s : X ^ W . Let K = s -1 (cl [/); then V is clopen in X , since cl U is clopen in W , an d from the construction of W we clearly have U ^ K ^ c l U . So V is the closure of U ; hence every open subset of X has clopen closure, i.e. X is extrem ally discon­ nected.

□

3.9 Since the projective objects in KHausSp all lie in the sub­ category Stone of Stone spaces, it is easy to see th at they are exactly the projective objects in Stone. (The argum ent of Lem m a 3.3 can be used to show that epimorphisms in Stone are surjective.) Applying Stone duality to this result, we get the *Corollary The injective objects in Bool are precisely the complete Boolean

algebras.

□

Just as for spaces, this result can be extended to a larger category than the one for which we originally conceived it: *Proposition The injective objects in DLat are precisely the complete Boolean

algebras. Proof Recall th at in Exercise II 4.5 we gave an explicit description of the left adjoint L of the forgetful functor Bool-»DLat, which enabled us to show that the unit m ap A ^ L ( A ) is a m onom orphism for any distributive

106

I II: Com pact H ausdorff spaces

lattice A. If A is injective in DLat, this m onom orphism m ust split; so A is a retract of L(A) in DLat, and hence a Boolean algebra. The C orollary above now shows it m ust be complete. Conversely, let A be a complete Boolean algebra. To show th a t A is injective in DLat, it suffices to show that the functor L preserves m onom orphism s; for then if we are given a diagram

9 '> A

in DLat with / m ono, we can replace it by the corresponding diagram L(B)------ MU ---- ^L(C) 9

A

in Bool and use the Corollary above. N ow since we know th at m onom orphism s in Bool correspond to surjective m aps of Stone spaces, it follows from the description of L given in II 4.5 th at we need to show th at a m onom orphism / :B -» C in DLat induces a surjection spec C-»spec B, i.e. that every prime filter of B is the inverse image under / of some prime filter of C. Let F be a prim e filter of B, and let I denote its com plem entary prime ideal; then the sets F = {ceC\(3beF)(f(b)^c)}

and r = {ceC\(3beI)(c^f(b))}

are respectively a filter and an ideal of C, and they are disjoint since / is one-to-one. So by Lemma I 2.3 and Theorem I 2.4 we can enlarge F' to a prim e filter F ” disjoint from /'; but then / “ 1{F") contains F and is disjoint from /, and so m ust equal F. So the result is proved.

D

N ote that the argum ent about extending prime filters, which we used in the proof above, was actually a special case of the result we were trying

Gleason's Theorem

107

to prove, namely the fact that the complete Boolean algebra 2 is injective in DLat.

f3.10 As well as identifying the projective objects in KHausSp, G leason also showed th a t every object of this category has a projective cover; i.e. for every X there is a surjection e : y X - > X where y X is projective. M oreover, if we dem and th at e be a m inim al surjection, then y X is determ ined up to (unique) hom eom orphism over X. O nce again, this result can be extended to m uch larger categories of spaces (or of locales) th an KHausSp, provided we insert the extra hypothesis th at e is proper. In this paragraph we shall give the construction ofyX for com pact H ausdorff X , and then sketch how it may be extended to arbitrary H ausdorff X ; for further generalizations the reader should consult the references listed in the Notes. F o r the tim e being, therefore, let X be a com pact H ausdorff space, and consider the Stone space yX = sp e c (Q -|“iW)* Since Q -|-,(X ) is a com plete Boolean algebra, y X is extrem ally disconnected. (Incidentally, we note in passing that y X could equivalently be defined as m in (Q(X)), since Exercise II 4.9 implies th at this space is hom eom orphic to spec (Q-^-1(Ar)).) If F is a point of y X (i.e. a prim e filter in f i - j ^ A -)), we say x g X is a limit of F if every regular open neighbourhood of x is a m em ber of F ; then as in P roposition 2.2 it is easy to show th at every F e y X has a unique limit e(F), an d in fact {e(F)} = f ] { c 1 V \ V e F}. From this inform ation, we may show as in 2.3 th at for every open U £ X we have e- HU)=^{F e y X \ ( 3 V e Q ^ ( X ) ) ( V

in Q(X) and V e F ) J,

so th at e is continuous. And for any x g X , the filter of regular open neighbourhoods of x can be extended to a prim e filter F, which clearly satisfies e ( F ) = x ; so e is sur­ jective. T o see th at e is a m inim al surjection, it suffices to show th at for any nonem pty basic open set U X in KHausSp whose dom ain is extrem ally disconnected; and e is charac­ terized up to unique isom orphism in the category KHausSp/X by these properties. □ W hat happens if X is H ausdorff but n o t com pact? In this case there will be prime filters in Q-i-i(X) which have no lim it; but each prim e filter will still have a t m ost one limit, and so we m ight try defining y X to be the subspace of spec ( Q ^ - [ X ) ) consisting of those prim e filters which have limits. Provided X is regular, the descrip­ tion of e ~1(t/) which we gave above is still valid, and so e : y X - * X is continuous. Also, it is easy to see th a t any nonem pty basic open set in spec (Q-]-|(X)) contains a convergent prim e filter, i.e. y X is dense in spec (Q-i-,(X)). So in view of Exercise Show th at a dense sublocale of an extrem ally disconnected locale is extrem ­ ally disconnected, we know that y X is extrem ally disconnected. The proof th at e is a minim al surjection is exactly as before; b u t this tim e we have also to show th at e is proper, which involves som e extra work. The proof of uniqueness then proceeds as in the com pact case. If X is H ausdorff but not regular, we have the further com plication th at the m ap e : y X - > X (as defined above) is n o t continuous. So in this case we have to equip the set of convergent prime filters with a finer topology th an the subspace topology (in fact it suffices to take the topology generated by the sets ^ " 1(C7), U open in X , to ­ gether with the open sets in the subspace topology), and thus there is further work to be done even in proving th at y X is extrem ally disconnected. We shall not give the details here.

f 3 . l l By restricting Theorem 3.10 to the category of Stone spaces and applying Stone duality, we obtain a result ab o u t Boolean algebras: *Corollary

Any Boolean algebra has a m inim al com pletion; i.e. for any Boolean algebra A there is a com plete Boolean algebra M{A) and a m onom orphism A - * M ( A \ such th at every m onom orphism from A to a com plete Boolean algebra B can be extended to a m onom orphism of Boolean algebras M(A)-*B.

□

Gleason's Theorem

109

N ote th a t we do n o t claim th at the extended m ap M(A)->B is a m orphism of com plete B oolean algebras; i.e. it need not preserve infinite joins and meets. If it were possible to achieve this extra condition, then we should have constructed a left adjoint for the inclusion CBool-^Bool - which we know to be impossible, by P ro p ­ osition I 4.10. (In fact it can be shown th at A t-*M(A) is left adjoint to the inclusion CBool->Boolc, where Boolc is the category of all Boolean algebras and ‘com plete hom om orphism s’ between them , i.e. m aps preserving all those joins and m eets which happen to exist.) T he elem ents of M{A) are of course the regular open sets in spec A, i.e. the “ 1~~1stable elem ents of the locale Idl(/1). N ow for an ideal I ^ A , we have ~ \ I = { a € v4|(V i g /)(flA / = 0 )J: so an elem ent a is in “ 1/ iff ~\a is an upper bound for I. Thus the condition / = I 1/ is equivalent to saying th a t every lower bound for the set of upper bounds for / is a m em ber of /. This enables us to generalize the construction of M(A) to arbitrary lattices (indeed, to arbitrary posets) A. If A is a poset and S £ A , we shall write w(S), l(S) respectively for the set of upper bounds for S and the set of lower bounds. W e define a cut in A to be a pair of subsets (L, U) such th at L=l {U) and U = w(L); and we w rite M(A) for the set of all cuts in A , ordered by ( L u U i ) ^ ( L 2, U 2) iff L l ^ L 2 (or equivalently Uy 2 l / 2). (Note th a t if (L, U) is a cut in a lattice A , then L is necessarily an ideal o f A and U is a filter; so this agrees w ith our previous definition of M(A) for a Boolean algebra A.) F o r every elem ent a s A, th e pair (j(a), |(a)) is a cut in A, which we denote by m(a)\ note th at a cut (L, U) is of this form iff L n U is nonem pty. Theorem F o r any poset A, M(A) is a com plete lattice, and the em beddingm : A->M(A) preserves all jo ins and meets which exist in A. Proof Let S be a subset of M(A). C onsider the set L 0 = f i[ L |( L , U ) e S } \ since L 0 ^ L for every (L, U )g S , we have u(L0) ^ U for every (L, U ) e S . and hence /(«(L0) ) s f ) {L\(L, U ) e S } = L 0. So (L0, u(L0)) is a cut in A> and it is clearly the greatest low er bound of S. Joins in M(A) are constructed similarly. N ow if a — A S in A. then we have j(a )= H U (s)ls e 5}; so m preserves all such meets, and similarly it preserves joins. □ T he lattice M(A) is called the MacNeille completion of A. It is interesting to com ­ pare the em bedding m : A->M(A) with the em bedding v4—► ldl(v4) defined using principal ideals; although the latter preserves all m eets which exist in A, it does not preserve any infinite joins. O n the other hand, we know th at Idl(/1) is a distributive lattice whenever A is (C orollary II 2.11), whereas distributivity is not in general inherited by the M acNeille com pletion, as the following example shows:

110

III: C om pact H ausdorjf spaces

Example Let X be an infinite set. P artition X into three disjoint infinite subsets X i , X 2 and X 3; let y 2, y3, ...) and ( z u z 2, z3, .. .)be two disjoint infinite sequences of elem ents of X 3, and define Yn= X 3 —{ y u . . . , yn}, Z n= X 3 — {z l, . . . , z„} for each positive integer n. (Note th at since the sequences (y„) and (zn) are disjoint, we have Ymu Z„ = X 3 for all m and n.) Let A be the sublattice of P X generated by the following sets: all singletons {x} for x g X ( U l 2, all sets of the form X y u Ym or X 2u Z m, and the set X 3. Since A is a sublattice of PX , it is clearly distributive. Suppose we have an em bedding f : A - * B 9w here B is a com plete lattice, preserving all joins and meets which exist in A. C onsider the following elem ents of B: b i = V {/({x})|xe X,}, b 2 = V {/({x}) | x e X 2}, B

a n d b 3= / ( X 3).

B

F irst we note th at since Ym\ j Z n= X 3 for all m and n, the only m em ber of A contain­ ing X ! u X 2 is X itself. Hence X = V !{x }|x g X j u X 2},

A

an d since / preserves this join, we have

1b = A X ) = V { / ( { x } ) | x e X , u X 2}= b, v b 2. B

N ow for every x g X , a n d every m > 0 , we have

{x )sx ,u y „ so th at / ( { x j l ^ / f X j U 1^) and hence by ^ f ( X y u Ym). So by A ^ / ( X , u y j A / ( X 3) = / ( y j ; b u t it is easy to see th at the only m em ber of A contained in every Ym is 0 , and so A A { Ym|m > 0 } = 0 . Since this m eet is preserved by f, it follows th at by a ^ 3 = 0b. But since X 3 £ 0 and / is one-to-one; so we deduce th at b 3^ b y and hence v b s + by. N ow consider b2 a (by v b 3). By an argum ent like th a t above, we can show th at b 2^ f ( X 2'uZ„) for every n< and by v b ^ ^ f ( X y ' u X ?l)', so b2 A(by v b ^ M X ^ Z J n l X y u X 3) ) = /( Z n). But once again, we have 0 ~ A a [Z „|n>0}, and so we deduce b2 A(by v b 3) = 0 B. N ow we have (by v b 2)A(by v b i ) = \

(by

a

(by

v b

a

3) ) v ( b 2 a (by

{by v b 3)=by v b 3; v f e 3)) = fe1 vQ

but

= by=f=by

v b 3.

So B is not distributive. Thus we have shown th a t A cannot be em bedded in a distribu­ tive com plete lattice in such a way as to preserve all joins and meets which exist in A ; in particular, M(A) cannot be distributive. In co n trast to the above example, we saw a t the beginning of this paragraph th at the property of being a Boolean algebra is inherited by the M acNeille com pletion. There is a sim ilar positive result for H eyting algebras, which we leave as an

V ietoris locales

111

Exercise Show th at M(A) is a H eyting algebra whenever A is. [G iven cuts (L t, U ^ and (L2, U 2) in M ( A \ define ( L u L/1)->(L2, U 2) = (L3, l / 3), where L 3= [a g v4|(Vb e Ly)(a A b e L 2)} and L/3 = m(L 3). T o prove L 3 = / ( l / 3), first show th at (b-K ‘) G l / 3 for all / j g L j and c g l / 2. Alternatively, show th at the m ap I \->l(u(I)) is a nucleus on Idl (/I), and deduce th a t M(A) is isom orphic to a sublocale of Idl(/1).]

4. Vietoris locales f4.1 This entire section represents a digression from our m ain line of progress. Its aim is to introduce a construction on com pact regular locales which we shall n o t m eet again until C hapter VII, where it will be exploited and subsequently generalized. In the context of locales, the construction appears to be new; but for spaces it has a substantial history. We begin by reviewing some of th a t history. It was observed by F. H ausdorff th at if (X, d) is a m etric space, one may define a m etric on the set K {X) of closed bounded subsets of X by setting d{A, B ) = max [ s u p ^ infbeBd(a, b \ su p ^ g intaeA d(a, b)}. (There are som e problem s in interpreting this definition if either A or B is em pty; the norm al practice - which we shall not follow - is to exclude the empty set from m em bership of K(X).) The space X is isom etrically em bedded in K ( X ) by the m ap xh»{x[. The H ausdorff m etric has its m ost pleasing properties when X is com pact (so th at K{X) consists of all closed subsets of X); in this case it can be shown th at K ( X ) is also com pact. T he latter result was generalized from m etric to topological spaces by L. Vietoris. He showed th a t for any com pact H ausdorff space X, there is a com pact H ausdorff topology on the set K{X) of (nonempty) closed subsets of X, which coincides with th at induced by the H ausdorff m etric in the case when the topology on X is induced by a metric. It turns out th at this space (the Vietoris space o r hyperspace of X) in­ herits m any interesting topological properties from X, and the theory of hyperspaces has been extensively developed in recent years. O f course, we may regard the points of the Vietoris space as being the open subsets of X rather th an their closed com plem ents, and so view it as a topology on the set Q(X). It is then easily seen th a t the definition of the V ietoris topology may be phrased entirely in term s of the lattice structure of Q(X), which opens up the road to defining the Vietoris space of a com pact regular locale. (Of course, this gives us nothing new if we assum e the axiom of choice; b u t one of our aim s is, as always, to elim inate the use of th at axiom wherever possible.) Nevertheless, there is an unsatisfying asym ­ m etry ab o u t this construction: starting from a locale, one ends up with a Vietoris space. O u r first aim is thus to define the Vietoris locale of a com pact regular locale; we shall then show th at its space of points is the Vietoris space as usually constructed.

112

I II: C om pact H ausdorjf spaces

t4.2 W e recall th at a subbasis for the V ietoris topology on K(X), X a com pact H ausdorff space, is given by the sets t(U)— {F

g

K(X)\F£ l / }

and m ( U ) ~ {F

g

K ( X ) \ F n U =^0}

where U ranges over all open subsets of X. If we replace the points of K(X) by their open com plem ents, these definitions become t ( U ) ~ { V e Q , ( X ) \ U u V — X} and m{U)={VeQ.(X)\U £V} and we take these definitions, with the obvious notational m odifications, for the subbasic open sets f(a), m(a) in the Vietoris topology on a com pact regular locale A. Lemma F o r any com pact regular locale A, the sets t(a) and m(a) satisfy the following identities: (i) t ( \ A) = A, t(a)nt (b) =t (a

Ab)

(ii) m(a) n m(a A b ) = m(a a b) (iii) t{a) n m{b) = t{a) n m(a a b) (iv) m( VS)= (J{m(s)|s e S } (including m{OA) = 0 ) (v) t( V S )= (J{r(s)|s g 5 } w henever S is directed (v i)

t(a v b ) —t{a)u{t(a v b)nm(b))

Proof M ost parts of this are trivial; we com m ent on three of them. (iii) If c g t{a)nm( b\ then (a Ab) v c = (a v c) A(b v c ) = b v c which is strictly larger th an b, and hence larger than a A b . So c € m ( a A h ) \ but from (ii) we have m(a a b) £m(b), so the result follows. (v) If c g t( V 5 ), then the elem ents c v s , s g S , form a directed family with jo in So by com pactness of A there exists s e S such th at c g t(s).

(vi) If c g t(a v b ) and c $ t(a), then a v b v c ^ a v c and so bvc=^c, i.e. c So t(a v b) £ t(a)u m (b).

g

1 A.

m(b). □

Exercise (i) If the topology on X is induced by a m etric d, show th at the Vietoris topology on K ( X ) is indeed induced by the H ausdorff m etric as defined in 4.1, W hat happens to the point 0 of K ( X ) in this topology? (ii) F o r A a com pact regular locale, a e A, define s(a )=

{b

g

A\a

^

b }.

Show th at s(a)=t(~\a) and th a t t(a)= (J{s(b)|a v b = 1}, and deduce th at the Vietoris topology on A is generated by the sets s(a) and m(a), a € A.

V ietoris locales

113

(iii) Show th at the topology on A generated by the sets s{a) (equivalently, by the t{a)) is exactly the Scott topology (II 1.9) of the poset A. (Note th a t the topology generated by the m{a) is the lower interval topology on A , i.e. the upper interval topology (II 1.8) of /lop.)

f4.3 We now define the V ietoris locale V(A) of a com pact regular locale A , essentially by taking the identities of Lem m a 4.2 as definitions. T hat is, we now regard the t{a) and m(a) as abstract sym bols rath er th an subsets of A, and we define L{A) to be the a-sem ilattice generated by these sym bols subject to the relations (i) f(U) = 1lm)> t ( a ) A t ( b ) = t ( a A b ) , (ii) m(a) a m(a a b) = m(a a b \ and (iii) t(a) a m(b) = t(a) a m(a a b). It is easy to see that each elem ent of L(A) has a unique expression in the form t{a) Am{by) Am{b2) a ■■■ a m{b„) (n^O ) where the b{ are pairwise incom parable (i.e. b i ^ b j i t t i = j ) and each b ^ a . F or simplicity, we shall denote this elem ent by w{a; by, b 2, • ■•, b„). N ow we define a coverage C on L(A) by requiring the following sets to be elements of C{w{a; . . . , bn)): (iv) [w(a; b u . . . , b„_u s)\s e 5} whenever bn= V S (here the index n m ay be replaced by any / with 1 ^ n — 1); (v) {w(s; b i , . . . , b„)\s e S } whenever S is directed and V S — a ; and (vi) {w{a'\ by, . . . , bn\ w{a; b0, b u . . bn)} whenever a —a v b 0: and we define V(A) to be the locale of C-ideals of L(A). Proposition V(A) is a regular locale. Proof F or simplicity of notation, we shall identify elem ents of L(A) with the principal C-ideals they generate. We begin by show ing th at a ^ b in A implies t(a)^t(b) an d m(a)^m(b) in V(A): for the first, note th at ~ ~ \ a v b = \ A implies m(~\a) v t ( b ) ^ t ( \ A)= 1V(A), by (vi), and m (“ la) Af(a) = m(Oi4) Af(a) = 0KM) by (iii) and (iv). Similarly, we have t ( ~ \a )v m( b )= \ and f(“ la) A in(a)=0. Now since the set {a g A\a ^ b } is directed and has join b, conditions (iv) and (v) tell us that each element of L{A) of the form m(b) or t(b) is a jo in (in V(A)) of elem ents well inside itself. The stability of the ^ relation under finite meets, plus the infinite distributive law, allow us to deduce the same statem ent for an arbitrary elem ent w(a; b y , , bn) of L(A) (i.e. an arb itrary principal C-ideal); but since each elem ent of V(A) is a jo in of principal C-ideals, this is sufficient. □ Corollary There is a closed em bedding i : A-> V(A). Proof Define i*(r(a))—i*(m (a))=a. It is easy to verify th a t this definition respects the relations (iH iii), and so extends uniquely to a semilattice hom om or­

114

I II: C om pact H ausdorff spaces

phism L(A)->A. M oreover, it preserves covers of types (iv)-(vi), and so extends uniquely to a frame hom om orphism i* : V(A)-*A, i.e. a locale m ap i : A-*V(A)> It is clear from the definition th at /* is surjective, so i identifies A with a sublocale of V(A); since A is com pact and V(A) is regular, this sublocale m ust be closed by P roposition 1.2(iii). □ Exercise If a and b are com plem entary elem ents of A, show that t(a) and m(b) are com plem entary elements of V(A). Deduce th a t V(A) is zero-dim ensional if A is. t4 .4

Theorem V(A) is a com pact locale. Proof Let R be a directed subset of V{A). As a first approxim ation to the jo in of R in V ( A \ consider the union ( J R of all its m em bers. It is clear th at [ JR is a lower set, and it follows easily from directedness of R th a t [ j R is closed under finite covers of types (iv) and (vi). So we wish to ‘close’ it under directed covers of types (iv) an d (v). O u r experience with the proof of T ychonoff s theorem (1.7) m ight lead us to expect th at a transfinite iteration would be needed to attain this closure; but in fact the presence of regularity (which we did not assum e in 1.7) ensures th at a single step will suffice. Let T be a dow nward-closed subset of L ( A ), closed under finite covers of types (iv) and (vi), and define D ( T ) = { w ( a \ b „ ) \ w ( d \ b\, . . . , b ' „ ) e T for all a' ^ a , b\ , b’n ^.bn}. Using covers of types (iv) and (v), and the regularity of A , it is easy to see th at D(T) is contained in the C-ideal generated by T. W e shall show th at it is itself a C-ideal. F irst consider covers of type (iv): suppose bn= V S and for each s e S we have w(a; f r _ l5 s) g D( T). N ow for any b'n^ b n we have \ A=~\b'n v V S = ~\b'„ v V{f g y4|(3s g S)(t ^s)}, and so by com pactness there exists a finite set {t j , . . . , tk] with each rf^ s f for some s{ e S and b'„^.tl v ■■* v t k. But then we have w(a'; b \ , . . . , b'n- t j ) g T for all a' ^ a, b\ (1 — 1), and 1 Since T is closed under finite covers, we deduce w(a'; b'u . . . , b'n) e T, and hence w ( a \ b l, . . . , b „ - l, b n) e D ( T ) . The argum ent for covers of type (v) is sim ilar b u t easier, since we only have to consider directed sets S. F o r covers of type (vi), suppose a = a x v b 0 and we have w(ai ; b„)eD(T) and w(a\ b0, b l____ bn) e D(T). F or any a' w e can find and b'0 ^.b0 such that ~ \ a ' v a \ vb'0 = \ A and hence ar^ a \ v /j '0 , since the elements of the form a\ vfr'0 form a directed set with join a. So if h ' ^ b i (1 ^ i ^ n ) we have

w (a \

; b\ , . . . , b'n) e T, w(a' : b'0, b\, . . . , b ' n) e T , and hence w{a' ; b'u . . . , b'n) g T.

Hence w(a\ by, . . . , bn) e D(T).

115

V ietoris locales

Reverting to our directed set R ^ V { A ) , we thus have V VlA)R = D ( [ j R ). In p ar­ ticular V R = 1 V (A) iff ^ U A ) = t ( ^ A ) € D([jR). But since \ A ^ l A , this implies and hence som e m em ber of R m ust be 1V[A). So V(A) is com pact.

t(l^) g

[jR, □

f4.5 Proposition V is a functor KRegLoc-^KRegLoc, and the em bedding of C orollary 4.3 is a n atural transform ation from the identity functor to V Proof P roposition 4.3 and Theorem 4.4 show th at V is defined on objects. T o define it on maps, let / : A->B be a m ap of com pact regular locales, and define V(f)*(t(b))— t(f*b), V(f)*(m(b)) = m( f*b) for b e B. Since f * preserves finite meets, it is easy to verify th a t V ( f ) * extends to a sem ilattice hom om orphism L(B)->L(A); and since f * preserves joins, it further extends to a frame hom om orphism V(B)-> V{A). The functoriality of f ^ V ( f ) is clear from the definition, as is the com ­ m utativity of A --------------1 ------------- » B

i

i

vk--- ^--- .J, since a m ap into V(B) is determ ined by the effect of its inverse image on elem ents of the form t(b) an d m(b).

□

Exercise Show th at the form ulae fi*(t(a)) = t(t(a)), fi*(m(a)) = m(m(a)) define a m ap of locales p, : V ( V ( A ) ) ^ V ( A ) for any A, and th at (V> i, p) is a m onad on

KRegLoc. (We shall encounter the algebras for this m onad in section VII 3.) f4.6 As prom ised, we now turn our attention to the points of V(A). Proposition The space pt (K(/l)) is hom eom orphic to the Vietoris space of A , as defined in 4.2. Proof G iven c e A, we define a function p* with values in {0, 1} by pf(t(a))=\

iffavc^l^,

p*(m(a))=\

iffa^c.

The verification th at p* extends to a fram e hom om orphism V(A)->2 has in effect

116

I II: C om pact H ausdorff spaces

already been given in the proof of Lem m a 4.2. Conversely, given a point p : 2->V(A), define c — V { b e A\p*(m(b)) —0}. Since p* preserves joins, it follows from condition (iv) th at p*(m(c))=0, and hence p*(m(b))= 1 iff b ^ c . Also, if u v c = 1, then from t(a) v m (c )= \ V{A) we deduce p*(t(a)) v 0 = 1, i.e. p*(t(a)) = 1. But if a v c

1, then for any a' ^ a we have "1 a ' ^ c

(since " la ' v a = l ) , and so p*(m(~\a'))=l. N ow t(a') A m { ^ \ a ' ) = 0 VlA), so p*(t{a')) = Q for all a '^ a , and hence p*(t(a)) = 0 by condition (v). T hus we have shown th at the m aps p* and p* agree on all elem ents of the form t(a) or m{b), and so p = p c. T hus we have established a bijection between pt (K(/l)) and A ; and it is clear from the definition of p* th at the canonical m ap : K (/l)-»P(pt (K(/l))) sends the abstract symbols t(a) and m(b) to the subsets of A given those nam es in 4.2. So the topology on pt (K(/l)) is the Vietoris topology. □ * Corollary F o r every com pact regular locale A , the V ietoris topology on A is com pact and Hausdorff. Proof C om bine 4.3 and 4.4 with the fact that com pact regular locales are spatial (Proposition 1.10). □ Exercise G iven an element w(a\ b p . . . , b n) of L ( A \ show that either a A b j = Q for som e i, or there exists a point p of V{A) with p*(w(a\ b y, . . . , b„))= 1. Deduce (without using the axiom of choice) th at the canonical inclusion Q(pt (^(/ID j-^K f/l) is dense. f4.7 The identification of pt (K(/l)) in 4.6 allows us to transfer the results we already know about V(A) into continuity results about the Vietoris topology. We give two examples in this section. Lemma (i) Let A be a com pact regular locale, p a point of A. Then the inclusion / : A->V(A) carries p into the point pc of V(A), where c is the prim e element of A corresponding to p (cf. II 1.3). (ii) Let / : A->B be a m ap of com pact regular locales. T hen the function /* : A->B is continuous for the Vietoris topologies on A and B. Proof (i) F o r the point p, we have p*(a)= 1 iff a ^ c , so the com posite p*i* agrees with p f on all elem ents of V(A) of the form m(a). But in the proof of P roposition 4.6 we saw th at a point of V(A) is actually determ ined by its effect on elem ents of this form. (ii) W e need to show th a t the continuous m ap V ( f ) induces the function

on

V ietoris locales points of V(A). But if pc is a point of V(A) and b

117 g

B, then pt(V(f)*(m(b))=

pf{m{f*b)) = 1 iff f * b ^ c , which is equivalent to saying b ^ f * c . So the inverse images of V ( f ) ■pc an d agree on elem ents of the form m(b); as in the first part, this is sufficient to show they are equal. □ As a further stage of translation, we may now rephrase these results in term s of the space of closed subsets of a com pact H ausdorff space X : Corollary (i) Let X be a com pact H ausdorff space. T hen the m ap xh»{x} is a closed em bedding of X in the space K( X) of closed subsets of X , where the latter is given the Vietoris topology. (ii) Let f : X - > Y be a continuous m ap of com pact H ausdorff spaces. Then the m ap K ( f ) : K ( X ) - * K ( Y ) which sends a closed set F ^ X to its image under / is continuous for the V ietoris topologies. Proof (i) T he point x of X corresponds to the prim e elem ent X —{x} of Q(X), so this is a straightforw ard translation from p art (i) of the Lemma. (ii) Similarly, the m ap : Q(A')->Q(Y) sends an open set U to the largest open V £ Y with f * ( V ) £ l / , which is easily seen to be the interior of Y —{ f ( x ) \ x g X — U}. So the corresponding m ap on closed sets sends F to the closure of { /(x )|x g F }; but since / is a m ap of com pact H ausdorff spaces, this set is already closed. T hus we m ay deduce from part (ii) of the Lem m a th a t K ( f ) is continuous.

□

Exercise Let A and B be com pact regular locales. Show th a t there is a continuous m ap q : V(A) x lV ( B ) ^ V ( A x ,B) defined by

q*(m(I))= ((m(a), m(b))\(a, b) e I) and

q*(t(I))= ((t(a), t(b))\(a, b) e / ) where / is an ideal for the usual coverage on A x B and the right-hand side of each equation denotes ‘C-ideal in V ( A ) x V ( B ) generated b y . . . ’. By considering the effect of this m ap on points, show th at if X and Yare com pact H ausdorff spaces then the m ap K{ X) x K ( Y ) -> K( X x Y) which sends (F, G) to F x G is continuous for the Vietoris topologies. Deduce th at if a is a continuous binary operation on a com pact H ausdorff space X, then (F, G)h*{a(x, y)\x e F, y e G} is a continuous binary oper­ ation on K{X). f4.8 O ne further result about the V ietoris space which we shall require in section VII 1 is the fact th at it is a topological sem ilattice: th at is, the binary union m ap K(X) x K(A')-»-K(A') (or equivalently the binary intersection m ap Q(X) x Q(X) ->Q(X)) is continuous for the Vietoris topology. As before, we shall prove this result

118

I I I : Com pact H au sdorff spaces

by first constructing an appropriate m ap of locales and then showing th at it induces the desired function on points. Lemma F o r any

com pact

regular

locale

A,

there

is

a

locale

m ap

n : V(A) x , V(A)-*V(A) such th at n*(t(a)) is the C-ideal in V(A) x V(A) generated by (t(a), t(a)) an d n*(m(fr)) is sim ilarly generated by (m(b), 1) and (1, m(b)). Proof As usual, this is simply a m atter of checking th a t the relations (i)—(vi) are preserved by the above definitions. M ost of these are trivial; we com m ent only on (iii) and (vi). F o r (iii), suppose (x, y)en*(t(a))nn*(m(b)). Then either x = 0 or y = 0 (which cases are trivial) or we have bo th x ^ t( a ) and y ^ f(a ); and in addition we have one of x ^ m ( b ) and y ^ m(b). In either case we m ay deduce one of x ^ m(a a b) and ^ m ( f lA l) ) ; so (x, y) e n*{m(a a b)). Similarly, for (vi) suppose we have (x, y) € n*{t{a v b)). Again dism issing the trivial cases, we have both x ^ t(a v b) and y ^ t ( a vfr); so we can write x = x ' v x" with x '^ f(a ), x'f ^m(b), and y = y' v y" similarly. T hen we have (x \ / ) g n*(f(a)), and both (x, y") and (x", y) are in n*(m{b)), from which it is straightforw ard to show th a t (x, y) belongs to the jo in of these tw o C-ideals. □ Corollary (i) F o r any com pact regular locale A, the function a : A x A ^ A is con­ tinuous for the Vietoris topology. (ii) F o r any com pact H ausdorff space X , the function u \ K ( X ) x K ( X ) -+K(X) is continuous for the V ietoris topology. Proof (i) G iven points pc, pd of V ( A \ the corresponding point (pc, pd) of V(A) x jV(A) is defined by (Pc. Pd)*(I)=

v

{pf(a) Ap*(b)\(a, b) e I }

where / is a C-ideal in V(A) x V(A). In particular, we have (Po Pd)*n*(m(b)) = (p*(m(b)) a 1) v ( l a pj(m(b)))# which is 0 iff b oth b ^ c and b ^ d , i.e. iff piCAdf(m(b))= 0- As in Lem m a 4.7, this is sufficient to show that n *(pc, Pd) = p,CAd>(ii) is a straightforw ard translation of (i). □ In general the binary jo in m ap A x A - > A is not continuous for the Vietoris topology; indeed, as we shall see in section VII 1, there are good reasons why it can n o t be continuous for any com pact H ausdorff topology on A. It is possible to prove at the ‘localic level’ th at the m ap n : V(A) x , V(A)-> V(A) satisfies the identities of the theory of sem ilattices (I 1.3); th a t is, (V(A), n, p lA) is an internal semilattice in the category KRegLoc - and indeed the functor V can be regarded as taking values in the category of such sem ilattices and hom om orphism s between them. We om it the details.

N o te s on chapter I I I

119

Notes on chapter III Section 1 Separation axiom s for locales are considered in [D ow ker and Strauss 1972] and [Isbell 1972], am ong other references. D ow ker and Strauss took regularity as their basic separation property; Isbell introduced the strong H ausdorff axiom (1.3) and proved th at regular locales are strongly Hausdorff. The counter­ example in 1.5 is also due to Isbell, who showed that it was not strongly H ausdorff in [1972] an d th at it was n o t Tv in [1975a]. Two further ‘approxim ations’ to the H ausdorff property, called fitness and subfitness, were also introduced by Isbell [1972]; the former was further investigated by M acnab and Sim m ons [Sim m ons 1978]. In [1978a], Sim m ons succeeded in giving a purely lattice-theoretic condition which is equivalent for spatial locales to the H ausdorff property, b u t it is unclear w hat (if any) significance Sim m ons’ condition has in non-spatial locales. P roposition 1.6 appears in [D ow ker and Strauss 1977]. The Tychonoff theorem (1.7) was proved for locales by Ehresm ann [1957]; see also [P ap e rt 1964] and [D ow ker an d Strauss 1977]. However, all these authors used an argum ent which depends on the M axim al Ideal Theorem , for the assertion ‘A locale is com pact iff all its m axim al ideals are principal’. The choice-free proof of 1.7 in the text is due to Johnstone [1981c]. F or spaces, T ychonoffs theorem was first proved by Tychonoff (who else?) [1929] using the axiom of choice; Kelley [1950] showed th at the theorem for spaces actually implies the axiom of choice. Kelley’s proof m akes essential use of non-sober spaces (specifically, arbitrary sets with the cofinite topology), and it seems likely th a t any attem pt to deduce the full axiom of choice from a Tychonofftype theorem will involve non-sober spaces (doubtless this is related to the fact th at m axim al ideal spaces need not be sober). B ourbaki [ 1940] gave a proof of T ychonoffs theorem using ultrafilter convergence, from which it m ay easily be deduced th at the theorem for com pact Hausdorff spaces follows from the Prim e Ideal Theorem (see [Los and Ryll-Nardzewski 1951], [R ubin and Scott 1954]; we can obtain the same result by observing th at (Tychonoff for H ausdorff spaces) follows from our Theorem 1.7 and P roposition 1.10). But in fact (Tychonoff for H ausdorff spaces) implies the Stone R epresentation Theorem (in the form ‘The spectrum of a Boolean algebra is com pact’), and hence is equivalent to PIT. (To see this, note first that if B X is the free Boolean algebra generated by a set X , then spec B X is hom eom orphic (w ithout choice) to the product of X copies of the discrete tw o-point space, since a prim e filter of B X is com pletely specified by saying which o f the generators are in it. N ow any Boolean algebra A has a free presentation, i.e. a coequalizer diagram BY=zBX->A in Bool; this induces an equalizer diagram spec /l->spec B X ^ t s p z c BY, so spec A is (hom eom orphic to) a closed subspace of spec B X = 2X.) The observation in Exercise 1.8(i) is due to M. H. Stone [1937], who used the term ‘sem i-regular’ for spaces whose topologies satisfy the conclusion of this exercise. W alker ([1974], Exercise IE) attributes the approach to S tone-C ech com pactifications sketched in 1.8 to B. Banaschewski; see also [W alker 1976] and [M ac Lane 1971], p. 121.

120

I I I : C om pact H au sdorff spaces

In the proof of Lem m a 1.9, we used the full strength of the M IT ; however, since P ro p o sitio n 1.10 can be deduced from Theorem II 3.4 (Exercise 1.11 (ii)), it follows from P IT (and in fact is equivalent to it, since it implies th at Idl(B) is spatial for any B oolean algebra B). It is not clear w hether Lem m a 1.9 implies the M IT ; one might hope to prove this by showing th a t the locale Q(max B) constructed in Exercise II 3.5 is com pact (and nontrivial whenever B is), b u t unfortunately the points of this locale need not be m axim al ideals o f B, since m ax B need not be sober. Lem m a 1.11 was first observed by A. Joyal (unpublished) in the case when B is a closed bou n d ed sublocale of Q(R); see [Johnstone 1981],

Section 2 T he history of S tone-C ech com pactifications will be discussed at length in the notes on section IV 2. T he particular case considered in Lem m a 2.1 was know n to D ow ker and Strauss [1977]. T he description of topological properties in term s of ultrafilter convergence (e.g. P roposition 2.2) was initiated by C artan [1937, 1937a] and extensively pursued by B ourbaki [1940]; see also the ‘pseudotopological spaces’ of C hoquet [1948] and the ‘lim it spaces’ of K owalsky [1954] and Fischer [1959], where (ultra)filter convergence is taken as the prim itive notion. W hilst m any topological properties have very sim ple and elegant characterizations in term s of ultrafilters, this approach to topology has the disadvantage th at one is constantly appealing to the P IT to ensure th a t ‘enough’ ultrafilters exist; thus it is in a sense diam etrically opposed to o u r ‘localic’ approach. T heorem 2.4 is due to M anes [1967, 1969]; we have largely followed M anes’ original proof. P are [1971] gave a slightly different p ro o f based on his version of Beck’s Tripleability T heorem ; see also [M ac Lane 1971], p. 153. Edgar [1973] gave an o th er p ro o f which is m ore in the spirit of classical universal algebra; Sem adeni [1974] gave a m ore ‘purely topological’ proof, and G o n sh o r [1979] gave a ‘nonstan d ard ’ proof. Theorem 2.5 is also due to M anes [1969]; for a related result, see [L iber 1978].

Section 3 Projective and injective objects (3.1) were first extensively studied in m odule categories, in connection w ith the developm ent of hom ological algebra see [B aer 1940] and [E ckm ann and Schopf 1953], for example. T he realization of their usefulness in other categories cam e rather later. In C orollary 3.2, we used the axiom of choice to obtain the statem ent th at free algebras are regular projective; for m any familiar algebraic categories (e.g. groups, A belian groups) this statem ent actually implies the axiom of choice (see [Blass 1979]). Extrem ally disconnected spaces were introduced by Stone [1937a], who proved Lem m a 3.5. Theorem 3.7 is due to G leason [1958], w ho also extended it to locally com pact spaces as in Theorem 3.8. R ainw ater [1959] noted the connection between projective spaces and Stone-C ech com pactifications of discrete spaces (our P ro p ­ osition 3.4); Strauss [1967] noted th at G leason’s argum ent (essentially, o u r Lemmas 3.6 and 3.7) could be extended to m ore general spaces. O f course, this generalization requires th e correct form ulation of the concept o f ‘proper m ap’ ; in this context it is

N o te s on chapter III

121

interesting to com pare the three editions of B ourbaki’s Topologie Generate [1940, 1951, 1961]. [1940] does not m ention proper m aps; [1951] gives the definition X-*-Y is proper if f ~ x(K) is com pact for every com pact Y’ - a definition which is equivalent to ours only for locally com pact H ausdorff spaces, but which is still com m only used by algebraic topologists (e.g. Spanier [1966], D old [1972]). In [1961] there is a whole section on proper m aps, with a definition equivalent to ours and several other characterizations (including o u r Exercise 3.8). T he term perfect is often used for continuous m aps satisfying only conditions (i) and (ii) of 3.8; however, m any au th o rs restrict their attention to H ausdorff spaces, for which condition (iii) is redundant. T he problem of defining propriety for m aps of locales was solved by Johnstone [1979, 1981], who also proved the localic version of G leason’s Theorem ; however, th e definition requires a good deal of topos-theoretic m achinery, and so is beyond th e scope of this book. T he characterization of injectives in Bool (Corollary 3.9) was first obtained by Sikorski [1948]; see also [H alm os 1961], T he extension of this result to DLat (P roposition 3.9) is the w ork of Balbes [1967] and Banaschewski and Bruns [1968]. T he G leason cover (3.10) was introduced by G leason [1958] for com pact and locally com pact H ausdorff spaces, and subsequently extended to regular spaces [Flachsm eyer 1963, Strauss 1967], H ausdorff spaces [Iliadis 1963, Banaschewski 1967, 1971], T0-spaces [Btaszczyk 1974], and finally to arbitrary locales [Johnstone 1980a]. Dyckhoff [1972, 1976] has given a different construction of projective covers for arbitrary spaces, which has the advantage of being functorial on Sp (this fails for the G leason cover, see [H enriksen and Jerison 1965]), b u t does not in general yield a m inim al surjection. T he M acNeille com pletion (Theorem 3.11) was introduced by M acN eille [1936, 1937] as a generalization of D edekind’s construction [1872] of the real num bers from the ordered set of rationals. It was investigated from a categorical viewpoint by Banaschewski and Bruns [1967]. Exam ple 3.11 is due to Crawley [1962]; F unayam a [1944] had earlier constructed a distributive lattice A such th at M(A) is not distributive, even though A can be com pletely em bedded in a distributive com plete lattice.

Section 4 As indicated in paragraph 4.1, the H ausdorff m etric was intro­ duced by F. H ausdorff [1914], although the germ of the idea was present in an earlier paper by Pom peiu [1905], T he V ietoris topology was introduced by L. Vietoris [1922] in a paper which seems to have been ahead of its tim e; although a good deal of w ork on hyperspaces was done by the Polish school (e.g. [K uratow ski 1932], [M azurkiew icz 1932]) before the war, it was all restricted to the m etric case and used the H ausdorff m etric rather than the V ietoris topology. The latter was rediscovered by F rin k [1942] under the nam e ‘neighborhood topology’, and extensively studied by M ichael [1951] under the nam e ‘finite topology’. F o r a detailed account of the m odern theory of hyperspaces (almost exclusively in the m etric case), see [N adler 1978]; for a historical survey, see [M cA llister 1978].

122

I I I : C om pact H au sdorff spaces

Isbell [1964] m ade extensive use of hyperspaces in the study of uniform spaces; accordingly, when he introduced the concept o f uniform locale [1972], it was natural th at he should m ake use of hyperspaces of locales, although, as he adm itted (p. 6), ‘it is em barrassing th at I cannot define a hyper locale’. T he approach to hyperspaces presented in this section, which is due to the au th o r and has n o t been published before, does n o t really relieve Isbell’s em barrassm ent, since it covers only the com pact case; b u t it does give one grounds for hoping th a t som e analogous treatm ent m ay be possible for uniform locales. T he com pactness of V ietoris spaces is usually proved by an argum ent involving Alexander’s Subbase Lem m a [A lexander 1939], which in tu rn depends on Z orn’s Lemma. O u r approach to this result (C orollary 4.6) shows th at it follows from the PIT. Indeed, there is a converse result: the assertion ‘F o r every com pact regular locale A y the Vietoris space o f A is com pact’ implies (in view o f Exercise 4.6) th a t the Vietoris locale V(A) is spatial for every such A, and hence (by C orollary 4.3) th a t A itself is spatial. M ore particularly, the assertion T o r every com pact H ausdorff space X, the V ietoris space of X is com pact’ implies the Tychonoff theorem for H ausdorff spaces (and hence the PIT ); for if the X y, y € T, are com pact H ausdorff spaces, then \ \ y X y m ay be em bedded in an obvious way as a closed subspace of K ( X \ where X is the one-point com pactification o f the disjoint union of the X r (This argum ent was suggested to the au thor by J. R. Isbell.)

Further note on Section 1 Since the above notes were w ritten, the logical status o f L em m a 1.9 has been settled; it is in fact equivalent to the P IT , as are the statem ents ‘T he space o f po in ts o f any com pact locale is c o m p a c t’ an d ‘A n arb itra ry p ro d u ct o f com pact sober spaces is c o m p a c t’. T o prove these equiva­ lences, it is necessary to com bine results o f P. T. Jo h n sto n e (A lm ost m axim al ideals, Fund. Math. 123 (1984), 197-209) w ith those o f A. R. Blass (Prim e ideals yield alm ost m axim al ideals, Fund. Mat h. , to appear).

IV Continuous real-valued functions

1. Complete regularity and Urysohn’s Lemma

1.1 It should by now come as no surprise to the reader th a t we are going to regard the real num bers as a locale rather than a space. The basic idea of the following definition is that, assum ing the set of rationals as given, we wish to make the set of open intervals w ith rational endpoints into a site of definition for the open-set locale of U. Let Q + denote the totally ordered set obtained by adding a top element oo to Q, and let Q~ similarly denote Q u { - o o } . W e partially order QT x Q + by i

a

n

d

4

^ 4 ';

then O r x Q + is a meet-semilattice, with top elem ent ( —oo, oo) and (p, q) A (pV) = (max {p, p'}, min {q, q'}). T hroughout this section, we’ll write B for this semilattice. We now define a coverage C on B to consist of the following covers: {a) 0 e C(p, q) whenever p > q \ (b) {(p, r), (q , s)} e C(p , s) whenever p ^ q c r ^ s ; and (c) ftp', q')\p < p ' < q ' < q } e C(p, q) whenever p < q . We write L{U) for the locale of C-ideals of B . F o r simplicity of notation, we’ll identify (p, q ) s B with the principal C-ideal it generates (which consists of all (r, s) such th a t either p ^ r < s ^ q or r ^ s), at least when p < q . It will be convenient to have a description of the C-ideal generated by a lower set S ^ B . F o r such an S, let 0 (S )= {{p, q)|there exist finite sequences (p,, p 2, - - ., p») and (Qu

Qn) with (Pb Qi) e s for each /,

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IV'.Continuous real-valued functions

and D(S)={(p, q)\(p\ q ' ) e S whenever p < p < q ' < q } . Lemma F o r any lower set S e B , DO(S) is the C-ideal generated by S. P r oo f Clearly, O(S) is the closure of S under covers of type (b), and D(S) is the closure of S under covers of types (a) and (c) (the former because if p > q , the condition for (p, q) to be in D(S) is vacuous). So it suffices to prove that if S is closed under covers of type (b), so is D(S). But if (p, r) and (q, s) are in D(S) and p ^ q < r ^ s, we can find q \ r with q < q ' < r r < r ; and then for any p', s' with p < p' < s < s we have (p', r ) and (q\ s') in S. Hence (p\ s') is in S, and so (p, s) e D(S). □

1.2 Proposition (i) L{U) is regular. (ii) F or any (p, q ) e Q x d , the closed sublocale L[p, ^f] of L(U) com plem entary to ( —oo, p) v (q , oo) is compact. Pr oo f (i) Using coverings of types (a) and (b),it is easy to see that if P < P ' < q < q , then (p', q ' ) ^ ( p i q) in L(R). So the coverings of type (c) say precisely that every principal C-ideal is a join of C-ideals well inside itself, and hence L(U) is regular. (ii) C onsider a set X of C-ideals, each containing ( —oo, p) and (q , oo), such th at (JX generates B as a C-ideal. Then by Lem m a 1.1 we have B = D O (S \ and hence we have (p'q') e O(S) for some p', g' with — o o c p ' c p and q < q ' < co. But only finitely many elements of S, and hence only finitely m any elements of X , are involved in the proof th at (p', g") e O(S); and these finitely many elements together with ( —oo, p) and (q , oo) will then suffice to prove th at ( —oo, oo)eO(S). So there is a finite subset X 'e X whose union still generates B as a C-ideal; i.e. X ' still covers

L[p, q],

□

1.3 Proposition

pt (L(U)) is hom eom orphic to U. Pr oo f By II 2.11, points of L{U) correspond to meet-semilattice

C om plete regu larity and U rysohrfs Lem m a

125

hom om orphism s x : 2 which send covers in C to joins. Given such a map, define p < x to mean x(p, o o )= l and p > x to mean x( —oo, p ) = l. We shall show that the sets L = { p e Q |p < x } and C/ = { p e Q |p > x } define a Dedekind section of Q, i.e. a real number. First, L and U are nonem pty since x( —oo, oo)— 1 and hence (by (c)) there exist finite p and q with x(p, q)= 1, whence p e L and q e U . Next, L is a low er section since p e L and p ' < p implies x(p', oo)^x(p, oo)= 1, i.e. p’ e L ; and L has no greatest m em ber by another application of (c). Similarly, U is an upper section with no least member. L and U are dis­ joint, for if p e L and q e U then x(p,