Categorical structure of closure operators with ...

Mathematics and Its Applications

D. Dikranjan and w. Tholen Categorical Structure of Closure Operators

Kluwer Academic Publishers

Mathematics and Its Applications

Managing Editor.

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Categorical Structure of Closure Operators With Applications to Topology, Algebra and Discrete Mathematics

D. Dikranjan b ianment of Mathematics, &Wvenity of Udine,

U Pw Italy amd

W. Tholen A rw ient of Mathematics and Statistics, M! 1Jbiversity,

AM rock Ontario, OMM&

JRR ACADEMIC PUBLISHERS

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 0-7923-3772-7

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

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All Rights Reserved ®1995 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and __._.__._...._.__ __..L.... frn.w the ^,"aht r.wn,r

In memory of our parents

Nishan Dikranjan and Elise 0. Shoulian Bernhard J. Tholen and Hildegard M. Kennepohl

Table of Contents Preface

Introduction

xi

xiii

I. Preliminaries on Subobjects, Images, and Inverse Images 1.1 .M-subobjects

1

1.2 Inverse images are M-pullbacks 1.3 Review of pairs of adjoint maps

3

1.4 Adjointness of image and inverse image

5

1.5 The right M-factorization of a morphism

6

1.6 Constructing images from right M-factorizations 1.7 Stability properties of M-subobjects

7

1.8 M-subobjects of M-subobjects 1.9 When the subobjects form a large-complete lattice

12

3

9

14

1.10 The right M-factorization of a sink

16

1.11 The last word on least and last subobjects

19

Exercises

20

Notes

23

2. Basic Properties of Closure Operators 2.1 The categorical setting

24

2.2 The local definition of closure operator

25

2.3 Closed and dense subobjects

26

2.4 Idempotent and weakly hereditary closure operators

27

2.5 Minimal and hereditary closure operators

30

2.6 Grounded and additive closure operators

34

2.7 Productive closure operators

36

2.8 Restriction of closure operators to full subcategories

38

Exercises

39

Notes

43

3. Examples of Closure Operators 3.1 Kuratowski closure operator, Cech closure operator 3.2 Filter convergence spaces and Katetov closure operator 3.3 Sequential closure, b-closure, 0-closure, t-closure 3.4 Preradicals of R-modules and Abelian groups 3.5 Groups, rings and fields 3.6 Graphs and partially ordered sets 3.7 Directed-complete posets and Scott closure Exercises

Notes

4. Operations on Closure Operators 4.1 The lattice structure of all closure operators 4.2 Composition of closure operators 4.3 Cocomposition of closure operators 4.4 Closedness and density for (co)composites 4.5 Properties stable under meet or join 4.6 Idempotent hull and weakly hereditary core 4.7 Indiscrete operator, proper closure operators 4.8 Additive core 4.9 Fully additive core 4.10 Minimal core and hereditary hull 4.11 Productivity of idempotent closure operators Exercises

Notes

5. Closure Operators, Functors, Factorization Systems 5.1 Pointed endofunctors and prereflections 5.2 Closure operators are prereflections

5.3 Factorization systems

5.4 Recognizing classes of C-dense and C-closed subobjects 5.5 Closure operators versus M-preradicals 5.6 M-preradicals versus E-prereflections 5.7 (C, D)-continuous functors 5.8 Lifting closure operators along M-fibrations

5.9 Application to topological groups 5.10 Closure operators and CS-valued functors 5.11 Closure-structured categories, uniform spaces 5.12 Pointed modifications of closure operators 5.13 Closure operators and adjoint functors 5.14 External closure operators Exercises

Notes

6. Regular Closure Operators 6.1 A-epimorphisms and A-regular monomorphisms 6.2 A-epi closure and A-regular closure 6.3 Computing the A-regular closure for reflective A 6.4 The magic cube 6.5 Frolik's Lemma 6.6 The strong modification of a closure operator 6.7 Regular closure in pointed and additive categories 6.8 Clementino's Theorem 6.9 Regular closure for topological spaces 6.10 Pointed topological spaces Exercises

Notes

7. Subcategories Defined by Closure Operators 7.1 The Salbany correspondence 7.2 Two diagonal theorems 7.3 Essentially equivalent closure operators 7.4 Regular hull and essentially strong closure operators 7.5 Characterization of additive regular closure operators 7.6 The Pumplun-Rohrl correspondence 7.7 The maximal epi-preserving extension 7.8 Nabla subcategories 7.9 Companions of A(C) in topological categories Exercises

Notes

x

8. Epimorphisms and Coweipoweredness 8.1 Categorical preliminaries 8.2 Reflectivity and cowellpoweredness 8.3 Epimorphisms in subcategories of Top - a first summary 8.4 Projective closure operators and the categories Haus(P) 8.5 Cowellpowered subcategories of Top 8.6 Non-cowellpowered subcategories of Top 8.7 Quasi-uniform spaces 8.8 Topological groups 8.9 Epimorphisms and cowellpoweredness in algebra 8.10 The Frobenius closure operator of fields Exercises

Notes

9. Dense Maps and Pullback Stability 9.1 Hereditariness revisited 9.2 Initial and open morphisms 9.3 Modal closure operators 9.4 Barr's reflector 9.5 Total density Exercises

Notes

Bibliography Index of Definitions

Notation Index Tables of Results

Preface Our motivation for gathering the material for this book over a period of seven years

has been to unify and simplify ideas which appeared in a sizable number of research articles during the past two decades. More specifically, it has been our aim to provide the categorical foundations for extensive work that was published on the epimorphism- and cowellpoweredness problem, predominantly for categories of topological spaces. In doing so we found the categorical notion of closure operators interesting enough to be studied for its own sake, as it unifies and describes other significant mathematical notions and since it leads to a never-ending stream of examples and applications in all areas of mathematics. These are somewhat arbitrarily

restricted to topology, algebra and (a small part of) discrete mathematics in this book, although other areas, such as functional analysis, would provide an equally rich and interesting supply of examples. We also had to restrict the themes in our theoretical exposition. In spite of the fact that closure operators generalize the universal closure operations of abelian category theory and of topos- and sheaf theory, we chose to mention these aspects only en passant, in favour of the presentation of new results more closely related to our original intentions. We also needed to refrain from studying topological concepts, such as compactness, in the setting of an arbitrary closure-equipped category, although this topic appears prominently in the published literature involving closure operators. Readers of the book are expected to know the basic notions of category theory (such as functor, natural transformation, limit), although many standard notions are being recalled in the text or in the exercises. Some of the exercises should be considered part of the exposition of the general material and should therefore not be omitted, while others deal with specific applications and can be selected according to the Reader's background and interest. Each section contains at most one Theorem, one Proposition, one Lemma, one Corollary, and one set of Remarks and Examples, with very few exceptions. Hence "Proposition n.m" refers to the Proposition of Section n.m; in the exceptional case that Section n.m does contain a second proposition,

this will be labelled as Proposition* of n.m. Readers interested in new results on (non-)cowellpowered subcategories of topological spaces as presented in Chapter 8 might be able just to browse through Chapters 2,4,6,7 and still understand the material. We have, over the past seven years, benefitted from the interest in and advice on our work from many colleagues, including Jill Adamek, Alessandro Berarducci, Reinhard Borger, Francesca Cagliari, Gabriele Castellini, Maria Manuel Clementino, Eraldo Giuli, David Holgate, Jiirgen Koslowski, Hans-Peter Kiinzi, Bob Lowen, Sandra Mantovani, Jan Pelant, Nico Pumpkin, Jill Rosicky, Alberto Tonolo, Anna Tozzi, Vladimir Uspenskij and Stephen Watson. We also thank the institutions that made possible our joint work: the Bulgarian Academy of Sciences, the Natural Sciences and Engineering Council of Canada, York University, Fernuniversitat Hagen, the Universities of L'Aquila, Sydney, Trieste and Udine. Finally we thank Xiaomin Dong

Preface

xii

and Sandro Fusco for proof-reading early versions of the text, Angelo Montanari for his computer support, Luis Molina for his help in providing the typescript of chapters 1-5, Francis Borceux for letting us use his diagram package, and the publishers

for their patience with the authors, who still feel that they have just started their project but would probably never have finished it without the moral support of their best friends Giovanna D'Agostino, Alberto Policriti and Helene Massam. Udine and Toronto, February 1995

Introduction Closure operators (also closure operations, systems, functions, or relations) have been used intensively in Algebra (Birkhoff [1937], [1940], Pierce [1972]) and Topology (Kuratowski [1922], [1933], Cech [1937], [1966]). But their origins seem to go back to foundational work in Analysis by Moore [1909] and Riesz [1909] who both presented related notions at the "IV Congresso Internazionale dei Matematici" in Rome in 1908

(as was pointed out recently by Germano and Mazzanti [1991]). Early appearances

of closure operators are also to be found in Logic (see Hertz [1922] and Tarski [19291) before Birkhoffs book on Lattice Theory 119401 led to more concentrated investigations on the subject, particularly by Ward [1942], Monteiro and Ribeiro [1942], Ore [1943a, b], and Everett [1944].

Category Theory provides a variety of notions which expand on the latticetheoretic concept of closure operator most notably through the notion of reflective subcategory (see Reyd (19641, [1972], Kennison [1965], Herrlich [1968]), predecessors of which are present in the works of Samuel [1948], Bourbaki [1957], and Sonner [1963]. The notions of Grothendieck topology and Lawvere-Tierney topology (see

Johnstone [1977] and Mac Lane and Moerdijk [1992]) provide standard tools in Sheaf- and Topos Theory and are most conveniently described by particular closure operators. Both lattice-theoretic and categorical views of closure operators play an impor-

tant role in Theoretical Computer Science, again in a variety of ways. We mention only Scott's work [1972], [1982] which laid the foundations of domain theory, and we point to the vast literature on generalized functorial Tarski-type least-fixedpoint constructions (see, in particular, Wand [19791, Koubek-Reiterman [1979], Kelly [1980]).

The immediate aim of introducing closure operators is to describe conveniently the closure of a substructure with respect to a certain desirable additional property. Well-known examples are the (usual Kuratowski) closure of a subspace of a topological space, or the normal closure of a subgroup of a group, or the Scott closure in a directed-complete partially ordered set. Lattice theorists usually define a closure

operation c of a lattice L (with bottom element 0 ) to be a function c : L -. L which is

extensive (m < c(m))

monotone (m < n

c(m) < c(n))

idempotent (c(c(m)) = c(m)) and sometimes require c to be also grounded (c(0) = 0) additive (c(m V n) = c(m) V c(n))

From the categorical point of view, these systems of axioms turn out to be both insufficient and too restrictive. They ignore the important fact that, in the examples mentioned before, the closure operation c is available in each subobject lattice,

Introduction

xiv

and that every morphism f : X -r Y is continuous with respect to the closure operation:

f(cx(m)) 5 cy(f(m))

for every subobject m of X . Like Cech [1959] in his Topology book, we do not assume idempotency of a closure operation a priori. Therefore, when calling a

subobject m with c(m) = m to be c-closed, in general, we do not expect the closure c(m) of m to be c-closed. However, normally c-closedness may eventually be achieved by repeated (transfinite) application of c : m < c(m) < c2(m) < . . . < c°O (m) .

In an arbitrary category X with a suitable axiomatically defined notion of subobject, a (categorical) closure operator C is defined to be a family (cx)xex satisfying the properties of extension, monotonicity and continuity (see Dikranjan and Giuli [1987a], Dikranjan, Giuli and Tholen [1989]). The first five chapters of this book give a comprehensive introduction to the most important special properties and constructions involving closure operators. In addition to idempotency and additivity,

these include hereditariness (for subobjects m < y of X with y : Y --* X , the

closure of m in Y is obtained by intersecting its closure in X with y) and productivity ( c preserves direct product of subobjects). Closure operators may be ordered like subobjects, and the properties of closure operators that we are interested in are either stable under taking infima or suprema. This is the reason why each closure operator has, for instance, an idempoteni hull (which, in most cases, may be "computed" by an iterative process, as indicated above) and an additive core. Under the transition from C to its idempotent hull, say, other properties may or may not be preserved. Here, for instance, additivity survives the passage, but hereditariness does not in general. We examine all these properties and constructions carefully, both in terms of theory and of examples, taken predominantly from topology, algebra, and discrete mathematics. This enables us not only to detect common features and construction principles, but also to point to striking dissimilarities. For instance, with respect to the seemingly harmless condition of groundedness, one shows easily that in the category of topological spaces each non-trivial closure operator is grounded, whereas in the category of R -modules only trivial closure operators are grounded. Similarly, additivity is a common property for closure operators in topology but extremely restrictive for R-modules. While Chapters 1-4 keep the needed categorical apparatus limited, in Chapter 5 we give various functorial descriptions and constructions with closure operators which underline the naturality of the notion. First of all, a closure operator of a category X is nothing but a pointed endofuncior of the category of all subobjects of X. Iterations of this endofunctor as used in functorial fixed-point constructions lead to its idempotent hull (if they 'converge"). Closure operators may also be interpreted

as generalized factorization systems: a morphism f gets factored through the closure of its image,

X-'c(f(X))-Y.

But only if the closure operator is idempotent and if it satisfies a weak hereditariness property does one obtain a (dense, closed)-factorization. However, these two special

Introduction

xv

classes of morphisms enjoy the important properties of colimit- and limit stability even in the general case. In the category of R-modules, closure operators (and their properties) correspond to (the theory of) preradicals. Not only does this correspondence offer a rich supply of examples, but it is also extendable to our general context and turns out to be useful in non-Abelian structures as well. Hence preradicals offer a third (but more restrictive) interpretation of closure operators. Categories which come equipped with a fixed closure operator behave like (large) "spaces" the interaction of which is described by continuous functors. Similarly to the weak (=initial) topology, for any functor F : X -> Y and any closure operator

D of Y , one has a coarsest (=largest) closure operator C of X which makes F continuous; analogously for final structures. Hence closure operators may be pulled back or pushed along functors. In important cases (if F is a fibration, or if F is left- or right-adjoint) one obtains concrete construction procedures for closure operators defined this way. For example, in the category of topological groups we are able to establish closure operators very effectively by pulling back closure operators of both the categories of (discrete) groups and of topological spaces along the respective forgetful functors.

Chapters 6-9 are, generally spoken, devoted to the epimorphism problem, that is: to the characterization of those morphisms f : A -+ B of a category A which satisfy the cancellation property (u f = v f at = v) . Closely related to this is the question whether A is cowellpowered, this is: whether for every object A in A there is only a small set (not a proper class) of non-isomorphic epimorphisms with domain A . We shall mostly assume that A is a full subcategory of X , and our aim is to find an effectively defined closure operator C of X such that the epimorphisms of A are characterized as the C-dense morphisms in A. Two typical examples from topology illustrate this approach. In the category of Hausdorff spaces (spaces in which distinct points can be separated by disjoint open neighbourhoods), the epimorphisms are exactly described by the dense maps with respect.to the usual Kuratowski closure (i.e., maps whose image is closed in the codomain). Furthermore, since the size of a Hausdorff space Y containing a dense subspace X is bounded by 2"-d(x) , this category is cowellpowered. In the category of Urysohn spaces (spaces in which distinct points can be separated by disjoint closed neighbourhoods), the epimorphism- and cowellpoweredness problem is much harder. It seems natural to consider the so-called 0-closure of a subspace M of a topological space X first introduced by Velic"ko [19661, which is given by the

points x E X such that every closed neighbourhood of z meets M . Although -epimorphisms of the category of Urysohn spaces are not necessarily 9-dense, the 9-closure leads to the right track: they are characterized as the 9°°-dense maps, with B°° the unbounded transfinite iteration of 0 , that is the idempotent hull of 0 . Schroder [1983) constructed, for every cardinal sc , an Urysohn space Y," of cardinality ec which contains the space Q of rational numbers as a Q°°-dense subspace. Consequently, the category of Urysohn spaces is not cowellpowered. We begin our investigations on epimorphisms in Chapter 6 which deals with the regular closure operator regA induced by a subcategory A of X . It was introduced in a topological context by Salbany [1976), but the factorization it induces in A is exactly Isbell's [1966] dominion factorization which led him to characterize

xvi

Introduction

the epimorphisms of semigroups and other algebraic categories (see also Cagliari and Chicchese [1982]). Since within the category A , regA-dense means epimorphism and regA-closed means regular monomorphism, effective computational methods for regular closure operators need to be developed. Often this is achieved by providing

a closure operator C of X such that its idempotent hull coincides with regA, at least when restricted to A ; for instance 0 does this job for A = Ury . In other instances, one first needs to "modify C along cokernelpairs" before being able to reach the regular closure via the idempotent hull. An intensive study of the cokernelpair X +M X in X of the subobject m : M -* X with X E A is in fact the first step in tackling the epimorphism problem in any non-trivial situation. For "most" categories X it turns out that for m to be regular monomorphism of A , it is necessary and sufficient that X +M X already belongs to A , as explained by the Magic Cube Theorem (6.4) and Frolik's Lemma (6.5).

There are various ways of reversing the passage A reg-4 which are being studied in Chapter 7. For example, as Hausdorff spaces are characterized as the topological spaces X with closed diagonal AX in X x X , one can show in a fairly general categorical context that the objects of any regular-epireflective subcategory A of X are those objects X with regd-closed diagonal (cf. Giuli and HuSek [1986], Giuli, Mantovani and Tholen [1988].) Hence the passage that assigns to any closure operator C the so-called Delta-subcategory of objects with C-closed diagonal is of particular interest in the context of the epimorphism problem. It is used to characterize the additive regular closure operators, which are of particular interest for the epimorphism problem in subcategories of topological categories. In general, having a subcategory A of X, it is often advantageous to look for intermediate categories B such that the inclusion functor A B preserves epimorphisms. We present two good candidates for B and describe them in terms of closure operators which are intimately connected with the regular closure operators.

Chapter 8 contains a variety of known or new results on Haus of Hausdorff spaces These are large families of subcategories either containing Haus or being contained in Haus for which we present unified criteria and constructions for epimorphisms and (non-)cowellpoweredness. While every "reasonable" ranked category in algebra is cowellpowered (see Isbell [1966], Gabriel and Ulmer [1971], Adamek and Rosicky [1994]), the epimorphism

problem remains highly interesting. We concentrate our investigations on areas where closure operators are useful in deriving new results. For instance, Theorem 8.9 gives a complete description of subcategories of R-modules with surjective epimorphisms, and Theorem 8.10 provides a closure-theoretic description of epimorphisms in the category of fields. As indicated above, closure operators may be described by (generalized) factor-

ization systems. On the other hand, factorization systems (.6,M) with special stability properties of the class E characterize reflective subcategories and localizations (see Cassidy, Hebert and Kelly [1985], Borceux and Kelly [1987]), the latter of which are described in Topos Theory by Grothendieck- and by Lawvere-Tierney topologies. LT-topologies are simply idempotent and weakly hereditary closure operators whose dense subobjects are stable under pullback. We discuss them briefly

Introduction

xvii

in Chapter 9, concentrating on the Delta-subcategory which they induce. Under light assumptions on the category, one can effectively construct the reflector into the Delta-subcategory, and its epimorphisms are just the dense morphisms. Closure operators can be used to study topological concepts, such as separatedness, regularity, connectedness, and compactness, in abstract categories which are endowed with a closure operator (see in particular Manes [1974], Herrlich, Salicrup and Strecker [1987], Giuli [1991], Dikranjan and Giuli [1988b], [1989], [1991], Fay [1988], Castellini [1992], Clementino [1992], Fay and Walls [1994]). We emphasize that it is not the aim of this monograph to pursue these concepts to any extent. However, notions of separatedness and (dis-)connectedness appear throughout Chapters 6-9 to the extent to which they are of interest in conjunction with the epimorphism problem.