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Internal categories in Mal’cev categories. (English) Zbl 0937.18006

The main result of this article is that the category of internal categories and functors in a Mal’cev exact category is again a Mal’cev exact category (exactness here is in the sense of M. Barr). This result applies not only to categories of groups, rings, Lie algebras etc., but also to some non-varietal categories such as the duals of the categories of abelian groups and of compact Hausdorff spaces. A regular category \({\mathcal C}\) is a Mal’cev category iff every reflexive relation in \({\mathcal C}\) is in fact an equivalence relation. [For other formulations, see A. Carboni, J. Lambek and M. C. Pedicchio, “Diagram chasing in Mal’cev categories”, J. Pure Appl. Algebra 69, No. 3, 271-284 (1990; Zbl 0722.18005).] After some needed preliminaries on Mal’cev categories, the bulk of the paper is taken up with the proof of the main result.

MSC:

18C05 Equational categories
18D35 Structured objects in a category (MSC2010)
18E10 Abelian categories, Grothendieck categories

Citations:

Zbl 0722.18005
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