This is the second volume of the author’s three volume treatise on ‘Categorical algebra’ in which a truly heroic effort has been made gather together and distill into one coherent whole of roughly thirteen-hundred pages, what most specialists would broadly agree are the essential results of fifty years of intense research into ‘pure category theory’. This is by no means an easy task, for much of this work has been scattered in dozens, if not hundreds of books, monographs, papers, journal articles, conference proceedings, and even preprints and ‘personal communications’. Of course, its not all here: ‘homological’ and ‘homotopical algebra’ are the most notable omissions, but even here the necessary general categorical background of functors, natural transformations, limits, exact sequences, categories of fractions, etc., is covered in ample detail, and, with the exception of exact sequences, all in Volume 1, ‘Basic category theory’ (1994; Zbl 0803.18001).
Volume 2, under review here, is subtitled ‘Categories and structures’. It covers all of the basics of additive, abelian, regular and exact categories, treats categorical model theory in some detail and depth, and includes a very succinct introduction to the theory of monads and fibrations. In outline form it gives a surprising amount of the basics of symmetric monoidal and closed categories and those enriched over them, and includes, but only in passing mention, topological categories.
The first two chapters are devoted to those major types of categories, abelian and regular, which may be axiomatically singled out as possessing certain elementary types of finite limits and/or colimits which are related to each other by ‘exactness’ or ‘commutation properties’. They are of broad interest because as consequences of their axioms, numerous absolutely basic constructions and theorems are available in them which can then be seen as immediately applicable to wide classes of structured sets and mappings commonly studied in mathematics. Thus Chapter 1 covers abelian categories (and along the way, pre-additive and additive categories). It is a fitting beginning since it was this theory that really started it all as set-theoretic arguments about elements of kernels and images of linear mappings gradually came to be seen as expressible at first as properties of the linear mappings themselves and their set-theoretic universal factorization properties and, ultimately, just as limit and colimit properties of abstract ‘morphisms’ under ‘composition’, i.e., purely as properties of Eilenberg and MacLane’s, at the time, seemingly trivial notion of ‘categories’. Chapter 1 develops the basic theory all the way through localization and the embedding theorems of Freyd, Mitchell, and Lubkin (whose approach the author uses). As has been noted, in a precise sense which has an amusing irony, these latter theorems seem to turn the theory upside down, since they allow the verification of complicated abstract ‘diagram chases’ to be carried out using modules and, yes, their elements. The bonus, of course, is now that the results are true in contexts such as sheaves of abelian groups where ‘an element of’ may have lost any real meaning.
Chapter 2, Regular (and Barr-exact) categories appropriately follows: for, if, as Freyd remarked, ‘abelian’ (with a small ‘a’) just means ‘having a very nice structure’, then the ‘exact’ categories are precisely those which too have a ‘very nice structure’, but are ‘non-abelian’. It is in this context that the familiar ‘congruence-quotient-image’ factorizations of ‘universal algebra’ are possible with the ‘regular’ epimorphisms behaving as if they were surjective maps. Here a genuine ‘calculus of relations’ is available. The author has a serious pedagogical problem at this point, however, for one can only see just how nice these categories are by either developing the theory in such tedious detail that the reader is convinced that yes, it does behave essentially as it does in sets, or one can appeal to Barr’s embedding theorem which effectively places exact categories inside categories of sheaves of sets (Grothendieck topoi) where epimorphisms are always locally surjective and set-theoretic arguments often suffice. But sheaf theory is Volume 3 [Categories of sheaves (1994)]of the treatise. This notwithstanding, the author chooses to outline Barr’s theorem and using what certainly will appear to the naive reader as ordinary (if somewhat fastidious) set-theoretic reasoning, gives justification to his proofs in the calculus of relations by asserting that these are really being carried out in the as yet totally unexplained, ‘internal language and logic of the topos’. Frankly, this reviewer is not certain how to get around this problem either.
The second part of this volume is broadly concerned with what is now called ‘categorical model theory’. Again it begins with the beginning: Algebraic theories, Chapter 3, is devoted to Lawvere’s very elegant recasting of ‘universal algebra’ in functorial terms through the notion of an ‘algebraic theory’ which for him becomes a small category \({\mathcal T}\) whose objects are of the form \(X^n\), where \(X^n\) is required to be the categorical product of the object \(X\) with itself \(n\) times. A ‘model of an algebraic theory’ is just a product preserving functor on \({\mathcal T}\) with values in another category, e.g., the category of sets, and a ‘morphism of models’ is just a natural transformation of such product preserving functors. The exposition in this chapter is very carefully done with the relation between this ‘categorical approach to universal algebra’ and the probably more familiar ‘classical approach’ explained in detail, as is Lawvere’s ‘structure-semantics’ duality between the models of a theory and the theory itself (although the author does not use this terminology). Lawvere’s regular projective generator theorem characterizing those set-based categories which are categories of models of an algebraic theory (‘algebraic categories’) is given along with its ‘algebraic functor’ variants. The chapter ends with a discussion of ‘commutative theories’, ‘tensor products’ of theories, and ‘a glance at Morita duality’.
Chapter 4, Monads, at first glance might appear out of place here to the novice reader. A monad \(M\) (=‘triple’) on a category is simply an endofunctor on the category together with two natural transformations, its so-called ‘multiplication’ and ‘unit’, which satisfy axioms looking exactly like the axioms for a monoid with composition of functors replacing the cartesian product of \(M\) with itself. They arose in homological algebra as efficient methods for producing resolutions and every pair of adjoint functors \(M=UF\) on a category gives rise to one. A quite formal construction, the category of Eilenberg-Moore ‘algebras’ for the monad (just consisting of arrows from \(M(X)\) to \(X\) compatible with the multiplication and unit of \(M\)), has an obvious functor back to the original category and this functor has an almost equally obvious left adjoint which sends \(X\) to the multiplication evaluated at \(X\), the ‘free algebra’ on the object \(X\). The composition of these two functors gives back the monad \(M\) and the right adjoint ‘underlying object’ functor \(U\) of any pair of adjoint functors which does this factors through this category of ‘algebras’. If the factorization is an equivalence, \(U\) is said to be ‘monadic’ (=‘tripleable’). The surprise and relevance here is that it turns out that these formal ‘algebras’ really are algebras: over the category of sets the underlying object functor of any category of ‘universal algebras’ including those with infinitary operations is monadic as long as it has a left adjoint (the free algebra). Thus not only are groups and rings and modules etc. monadic but compact Hausdorff spaces and complete atomic Boolean algebras as well. Their entire structure is captured through the free algebra with its inclusion of generators and the concatenation multiplication of its words. All of these categories are (Barr-)exact with the finitary theories of Lawvere corresponding to the ‘monads with rank’ on \({\mathcal S}ets\). Quite remarkably, in this chapter the author not only makes these at first seemingly implausible facts seem reasonable but has managed to gather together in one chapter virtually all of the useful theorems on ‘monadicity criteria’ for functors as well as their associated lifting and stability properties. It is a truly useful and succinct compendium. He concludes the chapter with a brief glance at ‘descent theory for modules’ and explains its relevance here: ‘effective descent maps of commutative rings are exactly those whose associated module functors are (co)monadic’. This observation, in particular, as well as quite a bit of the real meat of this chapter is due exclusively to Jon Beck and it is one case where the attribution is undisputed. The author’s not always scrupulously followed policy of giving no attributions here does them both a disservice. Absent also here is any discussion of the ‘Beck-Chevalley condition’ which is satisfied in the example and is what makes it work. Unfortunately, any discussion of it is also missing from Chapter 8, another place where it would seem to have a natural home. It seems to be one of the few missing topics that most specialists would agree is as much ‘main stream’ as others which were included.
Chapter 5, Accessible categories, returns more directly to categorical model theory. Generalizing Lawvere’s pioneering approach, a ‘theory’ formally becomes a small category with some ‘specified (possibly empty) structure’ and a ‘model’ in \({\mathcal S}ets\) becomes a functor into \({\mathcal S}ets\) which ‘preserves’ the structure in some well defined sense. The aim here is to characterize internally those categories which are categories of models of some particular type of theory. Thus, replacing Lawvere’s ‘finite powers’ with ‘finite limits’, leads to the notion of a ‘locally finitely presentable’ category of models characterized by being cocomplete and having a set of generators, each of which is ‘finitely presentable’ in the sense that its associated covariant representable functor preserves filtered colimits. The corresponding replacement of ‘filtered colimits’ with \(\alpha\)-filtered colimits, where \(\alpha\) is some regular cardinal, leads to the Gabriel-Ulmer theory of ‘locally \(\alpha\)-presentable’ categories. Finally, the replacement of ‘limit preserving functor’ with ‘flat functor’ (i.e., a filtered colimit of representables) leads to the notion of an ‘accessible category’ and their ultimate characterization as categories of models of Isbell-Ehresmann ‘sketches’ (which are small categories with a distinguished set of cones and cocones which become limits and colimits in the models). Although this chapter is somewhat technical with its inevitable cardinality discussions of ‘degrees of accessibility’, it does offer a quick tour of this important part of category theory of particular interest in theoretical computer science and paves the way for the notion of ‘geometric theories’ and their ‘classifying topoi’ which logically come in the next volume of this ‘handbook’. Categorical model theory, per se, ends with this chapter.
Chapter 6 gives a brief introduction to ‘Enriched category theory’, in which the ‘hom-sets’ are ‘enriched’ in the sense that they take their values in another category \({\mathcal V}\) which, at a minimum, is only required to have a bi-functorial ‘multiplication’ or ‘tensor product’ \(A\otimes B\) defined for each pair \((A, B)\) of its objects. Viewed as an operation on the category, this tensor product is required to be associative, unitary, and commutative up to (specified) coherent isomorphisms, i.e., \({\mathcal V}\) is a ‘symmetric monoidal category’. Thus ‘\({\mathcal B}\) is a \({\mathcal V}\)-enriched category’ is defined as in ‘the hom-set definition of a category’, only here \(\operatorname{Hom}_{\mathcal B} (X,Y)\) is only an object of \({\mathcal V}\) and the \({\mathcal V}\)-enriched category’s ‘law of composition’ is given by a morphism \(\operatorname{Hom}_{\mathcal B} (X,Y) \otimes \operatorname{Hom}_{\mathcal B} (Y,Z)\to \operatorname{Hom}_{\mathcal B} (X,Z)\) in \({\mathcal V}\). If \({\mathcal V}\) is \({\mathcal S}ets\) with cartesian product as tensor, one recovers ordinary categories, but here the only ‘hom-sets’ necessarily involved are those of \({\mathcal V}\). A striking example of this was observed by Lawvere: the set \(\underline {\mathbb{R}}_+\) of extended positive real numbers is a category since it is a partially ordered set under the order relation and is a symmetric monoidal category under addition as tensor product. If \(X\) is a metric space then its distance function makes \(X\) into an \(\underline {\mathbb{R}}_+\)-enriched category. Chapter 6 also gives a brief introduction to symmetric monoidal closed categories where the tensor has a right adjoint. (These are very much a subject on their own, especially when the tensor is the categorical product, where they are called cartesian closed categories.) The basic definitions of enriched category theory including \({\mathcal V}\)-functors, ‘change of base’, enriched adjunctions, ‘weighted limits’ and general exponentiation (‘cotensors’) are all covered. He also discusses in this chapter the enriched version of volume 1’s theory of distributors. Among other things, in chapter 6 of that volume he proved that ‘a small category \({\mathcal A}\) is Cauchy complete’ (meaning ‘all idempotents in \({\mathcal A}\) split’) is equivalent to the assertion, ‘a distributor from 1 to \({\mathcal A}\) has a right adjoint if and only if it is isomorphic to a functor from 1 to \({\mathcal A}\)’. In long series of exercises (in fact, all but four of the entire set of the chapter) he outlines a proof that in the enriched category context for distributors, if one views a metric space \(M\) has an \(\underline\mathbb{R}_+\)-enriched category as in Lawvere’s example above, and notes that \(\underline\mathbb{R}_+\) is not only symmetric monoidal under addition, but both complete, cocomplete, and closed (with \([s,t]=\max\{s-t,0\}\) as cotensor), then the metric space \(M\) is Cauchy complete in the classical sense iff the identically phrased theorem of vol. 1 holds in the enriched sense; in short, ‘\(M\) is (classically) Cauchy complete iff it is \(\underline\mathbb{R}_+\)-enriched (categorically) Cauchy complete’. This is vol. 1’s long promised justification for using the term ‘Cauchy complete’ as an abbreviation for ‘all idempotents in \({\mathcal A}\) split’. Although the result (essentially due to Lawvere) is fascinating as mathematics, to use it as the justification for appending the august name of Cauchy to this simple (but extraordinarily important) categorical concept so tenuously related to Cauchy’s work, still just seems a bit lame. ‘Karoubi-complete’ or ‘Lawvere-complete’ would be more appropriate and a good opportunity has been missed to reestablish rational terminology.
The author’s selection of topics in this chapter does seem reasonable as an introduction to the domain of ‘braided monoidal categories’ so much advanced by the ‘Australian School’. But for the recent Joyal-Street work, more fully detailed texts such as G. M. Kelly’s ‘Basic concepts of enriched category theory’ [Lond. Math. Soc. Lect. Note Ser. 64 (1982; Zbl 0478.18005)]are available.
Chapter 7, Topological categories, is very brief, only somewhat over twenty pages. The aim of this part of category theory is to categorically capture the properties of categories of topological spaces and related functors. Since it is only in very rare instances (like compact Hausdorff spaces) that these categories are algebraic in any sensible use of the term, virtually none of the methods of categorical model theory are applicable here, and no sets of neatly packaged categorical axioms have yet appeared which seem at all satisfactory, at least nothing approaching the elegance of Barr’s notion of an ‘exact category’. Topological spaces are categorically so ‘ill behaved’ that the Grothendieck school formally proposed replacing spaces with the categories of set valued sheaves on them since this category anyway contained all of the principal homological invariants that they wished to study, and as a category, was almost as good as the category of sets. In fact, this was the reason for their calling such a category of sheaves a topos. Categories of spaces in the ordinary sense, nevertheless, are abundant, and furnish the category theorist with a rich source of examples and, even more abundantly, of counter-examples! The formal theory of ‘topological categories’, due principally to H. Herrlich and collaborators has produced one concept which is certainly enduring, the concept of a ‘topological functor’. Such functors capture categorically Bourbaki’s set based concept of initial and final structures (\(f\) is a morphism iff its composition with a given family of maps is a morphism), but rephrased as a condition on small cones or cocones of morphisms relative to some necessarily faithful ‘underlying functor’. Topological spaces to \({\mathcal S}ets\) is still the canonical example. Although the literature on topological categories is voluminous, this is the only part of the theory discussed by the author in any depth. Indeed, most of this short chapter is devoted to a discussion of the eternal quest for a ‘good category’ of topological spaces, with ‘good’ being defined by the categorically minded, and ‘cartesian closed’ being a principal requirement. The usual suspects are rounded up and the usual culprit of ‘compactly generated spaces’ is apprehended.
The final Chapter 8, Fibered categories, is an attempt to both give a definitive elementary account of Grothendieck’s original categorical theory of ‘fibrations’ and Benabou’s novel idea of giving them an absolutely central place in foundations. This is an important topic and some more accessible source of information on ‘fibrations’ is sorely needed since confusion in the literature (both terminological and just plain logical) abounds. For the most part, the author presents the elements of the theory in much the same way as A. Grothendieck [Catégories fibrées et descente, in: SGA1, Lect. Notes Math. 224, Exposé 6 (1971; Zbl 0234.14002)]but unfortunately, the real motivation for their study seems misplaced. For Grothendieck, the 2-equivalence of the category of fibrations above a given category \({\mathcal E}\) with the category of contravariant pseudofunctors on \({\mathcal E}\) (the ‘Grothendieck construction’) meant that ‘sheaf theory’ and ‘descent theory’ could be placed on an even footing without the baggage of the coherent isomorhisms of pseudofunctors. Pseudofunctors abound, any place that one has ‘pull-backs’, for instance. That this ‘baggage’ cannot really be ignored or ‘chosen once and for all’ and still remain in the same category [roughly, the approach of the theory of ‘indexed categories’ of R. Paré and D. Schumacher, Abstract families and the adjoint functor theorems, in: Indexed Categ. Appl., Lect. Notes Math. 661, 1-125 (1978; Zbl 0389.18002)]can be made clear from the fact that a homomorphism of groups, viewed as a functor of one-object categories, is a fibration iff it is surjective. A choice of a set-theoretic section (a transversal) is then a ‘cleavage’ for the fibration, and the associated pseudofunctor is precisely the associated ‘Schreier factor system’ for the extension. The 2-equivalence cited above is the correspondence of factor systems and group extensions. Fibrations are ‘split’ iff a functorial cleavage can be chosen, in which case the corresponding pseudofunctor is a functor. But not all group extensions are split. This clarifying example is not mentioned by Grothendieck, but his choice of terminology makes it clear that he was aware of the problem. The problem solved by Grothendieck through ‘fibrations’ is analogous to imagining that somehow Schreier factor systems arose naturally all over the place (as do pseudofunctors) and finally someone discovered that they were just a way to describe the much simpler concept of a group extension.
Benabou had the novel idea to carry the central role that Grothendieck intended for fibrations one step further and proposed for them a central role in a new foundation for mathematics. He showed how the concept of families of objects of a category could very neatly be described via a fibration and illustrated how limits, completeness, and definability were always logically and inherently related to some base category via properties of fibrations. As far as the reviewer knows, this program has yet to be completed and all that the author presents are the written portions of it that had previously been almost only ‘privately circulated’. Still, one can be grateful for that and this chapter on fibrations does assemble in one place the notable work of a number of people, that of John Gray in particular. Like Grothendieck, they are not cited directly.
But all these criticisms are quibbles. This is a remarkable piece of work so far and deserves the mathematical community’s grateful thanks.