On categories with effective unions. (English) Zbl 0661.18002
Categorical algebra and its applications, Proc. 1st Conf., Louvain-la-Neuve/Belg. 1987, Lect. Notes Math. 1348, 19-35 (1988).
[For the entire collection see Zbl 0644.00009.]
This paper explores some interesting connections between topos theory [cf. P. T. Johnstone, Topos theory (1977; Zbl 0368.18001)] and the theory of abelian categories [P. Freyd, Abelian categories (1964; Zbl 0121.02103)]. The key is a new kind of exactness condition which allows for the treatment of many constructions in each theory, in a unified way. Examples of such constructions include Grothendieck’s theorem on the existence of injective cogenerators, and the exactness of right exact functors and torsion theories and topologies. The main condition – having “effective unions” – is inherited by slices, coslices, products, and disjoint unions of categories, as well as the formation of functor categories. The results are of particular interest coming from someone who is an expert in both fields.
This paper explores some interesting connections between topos theory [cf. P. T. Johnstone, Topos theory (1977; Zbl 0368.18001)] and the theory of abelian categories [P. Freyd, Abelian categories (1964; Zbl 0121.02103)]. The key is a new kind of exactness condition which allows for the treatment of many constructions in each theory, in a unified way. Examples of such constructions include Grothendieck’s theorem on the existence of injective cogenerators, and the exactness of right exact functors and torsion theories and topologies. The main condition – having “effective unions” – is inherited by slices, coslices, products, and disjoint unions of categories, as well as the formation of functor categories. The results are of particular interest coming from someone who is an expert in both fields.
Reviewer: Marta Bunge (Montreal /Quebec)
