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Accessible categories: The foundations of categorical model theory. (English) Zbl 0703.03042

Contemporary Mathematics, 104. Providence, RI: American Mathematical Society (AMS). viii, 176 p. $ 31.00 (1989).
Categorical model theory, like many young and rapidly-expanding subjects, suffers from uncertainty about where its frontiers should be drawn. In the Introduction to this book, the authors argue that it should properly be defined as the study by categorical methods of models of (infinitary) theories in the category of sets; it is thus more restricted than categorical logic, which was the subject of an earlier monograph by the first author and G. E. Reyes [First order categorical logic, Lect. Notes Math. 611 (1977; Zbl 0357.18002)], and which is concerned with the idea that theories themselves are categories (with suitable structure) and their models in appropriately-structured categories are structure- preserving functors. In the present book the theories-as-categories paradigm does not play an important role; what does figure prominently is the paradigm of theories-as-sketches.
The notion of a sketch was introduced by C. Ehresmann in the late 1960s, but was rather neglected by category-theorists (other than Ehresmann’s own students, notably C. Lair and R. Guitart) for many years; its first appearance in a book originating outside the “Ehresmann school” was in 1985, in “Toposes, triples and theories” by M. Barr and C. Wells (1985; Zbl 0567.18001). Intuitively, a sketch is a presentation of a category with structure (the latter being understood as certain “potential” limits and/or colimits), and its models are mappings into the category of sets which turn this “potential” structure into actual limits and colimits. It thus occupies an intermediate position between the (inductive) syntactic presentations of theories which are traditional in logic, and the full acceptance of theories as categories. The advantage of the notion is that sketches are easily manipulated by categorical techniques, thus providing an easy route to studying constructions of new theories from old.
The name “accesssible category” is due to A. Grothendieck, though the definition adopted here is slightly different from Grothendieck’s. A category is called accessible if, for some regular cardinal \(\kappa\), it has \(\kappa\)-filtered colimits and a small set of \(\kappa\)-presentable (strong) generators. The definition captures the categorical essence of the Löwenheim-Skolem theorem; and the key result which justifies the definition (and which is due originally to Lair, although it was rediscovered independently by the authors of the present volume) is that accessible categories are (up to equivalence) exactly the categories of models of sketches. There is also a close link with the locally presentable categories of P. Gabriel and F. Ulmer [Lokal präsentierbare Kategorien, Lect. Notes Math. 221 (1971; Zbl 0225.18004)]; the latter are exactly the complete (equivalently, cocomplete) accessible categories, and they can also be characterized (as Gabriel and Ulmer knew) as the categories of models of those sketches which contain only limit cones. One could say without much exaggeration that the book under review entirely supersedes the one by Gabriel and Ulmer, in that all the main results of the Gabriel-Ulmer book appear here in the more general setting of accessible categories.
But the book does more than this: one feature which distinguishes it from Gabriel-Ulmer is the fact that the authors take 2-categorical matters seriously. Thus, when they prove that the (meta-)2-category of accessible categories and accessible functors is closed under Limits in CAT, they mean Limits with a capital L (which is their appealing shorthand for the weighted (bi)limits that one needs to consider in a 2-category), and not just ordinary limits. Much of the book is devoted to studying the closure properties of the 2-category of accessible categories, and the way in which these reflect the constructions which one can carry out on sketches; it is certainly praiseworthy that the 2-categorical details of this study have not been skimped, although it makes the text rather heavy going in places.
Although, as mentioned earlier, the main thrust of the book is towards the study of categories of set-valued models of sketches, the authors do in their final chapter make contact with the wider concerns of categorical logic by studying the category of models of a sketch in an accessible category. Here, it has to be said that the situation is less pleasant than the corresponding one for locally presentable categories: such categories of models are not always accessible, and some work remains to be done on understanding the precise conditions under which they are. Overall, a very large proportion of the results in the book are either new, or else presented in readily accessible form (pardon the pun!) for the first time. The book is clearly destined to become a standard reference for this material, and the authors deserve our thanks for making it available.
Reviewer: P.T.Johnstone

MSC:

03G30 Categorical logic, topoi
03C75 Other infinitary logic
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
18-02 Research exposition (monographs, survey articles) pertaining to category theory
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