Some results on locally finitely presentable categories. (English) Zbl 0615.18002

Makkai, M.; Pitts, A. M.

Trans. Am. Math. Soc. 299, 473-496 (1987).

This paper gives, first, some variations on the definition of locally finitely presentable categories (L.F.P.) and their morphisms, and, on the duality with small categories with finite limits. Then it proves the following results:
Given an L.F.P. \({\mathcal B}\) and a functor \(F: {\mathcal A}\to {\mathcal B}\) faithful and full on isomorphism, if \({\mathcal A}\) has limits and filtered colimits preserved by F, then \({\mathcal A}\) is L.F.P., and F has a left adjoint. In particular, if \({\mathcal A}\) is a full subcategory of an L.F.P. \({\mathcal B}\), closed under limits and filtered colimits, then \({\mathcal A}\) is L.F.P.
Moreover, a characterization of those left exact functors \(I: {\mathcal C}\to {\mathcal D}\) between small categories with finite limits, for which the functor \(I^*: {\mathcal L}ex[{\mathcal D},{\mathcal S}et]\to {\mathcal L}ex[{\mathcal D},{\mathcal S}et]\) is full and faithful, is given. It provides a proof of the fact that, whenever a structure is represented as the structure of global sections of a sheaf, then the structure can always be constructed by using limits and filtered colimits on the stalks of the sheaf.

Reviewer: Y.Diers

MSC:

18B99	Special categories
18C10	Theories (e.g., algebraic theories), structure, and semantics
03G30	Categorical logic, topoi
18A35	Categories admitting limits (complete categories), functors preserving limits, completions

Keywords:

global sections of sheaf; locally finitely presentable categories; small categories with finite limits; structure

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