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Non-pointed abelian categories

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Arnaud Duvieusart, Sandra Mantovani, Andrea Montoli

We study a property (P) of pushouts of regular epimorphisms along monomorphisms in a regular context. We prove that (P) characterizes abelian categories among homological ones. In the non-pointed case, we show that (P) implies the normality (in the sense of Bourn) of all subobjects, that any protomodular category satisfying (P) is naturally Malâ€™tsev, and that an exact category is penessentially affine if and only if it is protomodular and satisfies (P). An example of such a category is the one whose objects are the abelian extensions over an object in a (strongly) semi-abelian category; by exploiting some observations in this context, we also provide a characterization of strongly semi-abelian categories by a variant of the axiom of normality of unions.

Keywords:
Abelian categories, Regular categories, Strongly semi-abelian categories

2020 MSC:
18E08, 18E10, 18E13

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 32, pp 1227-1248.

Published 2022-10-28.

http://www.tac.mta.ca/tac/volumes/38/32/38-32.pdf

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