Some Results on Locally Finitely Presentable Categories

We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors T. C —» D between small categories with finite limits for which the functor /*: LEX(D,Set) —> LEX(C,Set) induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site C = (C, J) on a category C with finite limits, defined by a subcanonical Grothendieck topology J, the closure in LEX(C, Set) under small limits and filtered colimits of the models of C is the whole of LEX(C,Set). Introduction. In this paper, we make contributions to the logic of 'essentially algebraic' theories. Although the terminology will be categorical, the motivation is a model-theoretical one: our interest lies in arriving at results connecting the syntax and the semantics of that logic. The definition of 'essentially algebraic' theory can be given conveniently only in categorical terms. As is usually done, we identify such a theory with a small category having (all) finite limits. The 2-category of all such categories, with functors preserving finite limits as morphisms (1-cells), and all natural transformations as 2-cells, is denoted by Lex; LEX denotes the 2-category with the smallness restriction removed. Set, the category of (small) sets, is the 'standard' 'theory', an object of LEX. A 'model' of a theory C G Lex is a morphism C —> Set in LEX; LEX(C, Set) is the 'category of models of C Talking about relations of syntax and semantics amounts, roughly, to relating C and LEX(C,Set). Following [KR, we could call our subject the "doctrine of Cartesian logic". The doctrine of Cartesian logic corresponds to a logic with existential quantification restricted to cases when unique existence is assured. For a detailed account of this correspondence, cf. [C]. The main fact of the correspondence can be stated simply. Call a first-order theory T over a language L axiomatized by sentences of the form Vx(y?(x) —> 3=1yi/>(x, y)) with D in Lex that induce a full and faithful morphism F*: LEX(D, Set) -> LEX(C,Set) in LFP. Although the two results concern closely related situations, their statements or their proofs look pretty unrelated. Another result of [V] (Proposition 16) is the fact that any full subcategory of the category of all L-structures (with homomorphisms as morphisms) for which the inclusion creates limits and filtered colimits is an elementary class. Lemma 2.2 in the present paper is a more general form of this result. Using 2.2, and material from §1 we prove (Corollary 2.4) that any full subcategory of a l.f.p. category for which the inclusion creates limits and filtered colimits is again l.f.p.; and even the more general statement where the inclusion is required to be full only on isomorphisms (Proposition 2.3). Volger's remarks after his characterization theorem (Theorem 14) seem to amount to a proof of the weaker result, Corollary 2.4 (although the result as such is not stated). However, as far as we can see, the proof is incomplete (it does not seem to follow that the categories of models of T and T*, in Volger's remarks, are equivalent). We shall give some comments as to why 2.3 is of interest beyond its having 2.4 . as a consequence. Finally, a result that can be deduced from the work of Volger is the fact that whenever a structure is represented as the structure of global sections of a sheaf, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LOCALLY FINITELY PRESENTABLE CATEGORIES 475 then the structure can always be constructed by using (possibly repeatedly) limits and filtered colimits on the stalks of the sheaf. The precise, and more general, statement is Theorem 3.3. Our proof of it uses two of the above-mentioned results: 2.4 and 2.6. An interesting-looking corollary concerning subcanonical Grothendieck topologies ends the paper. 1. Gabriel-Ulmer duality. Let LEX be the 2-category of all categories having finite limits; LFC the 2-category of all categories having all (small) limits and filtered colimits. The morphisms (1-cells) of LEX are all functors between categories with finite limits preserving finite limits; the 2-cells are all natural transformations between such functors; similarly for LFC. We have forgetful functors (inclusions)