Semi-abelian exact completions

Abstract

We characterize the categories which are projective covers of regular protomodular categories. Our result gives in particular a characterization of the categories with weak finite limits with the property that their exact completions are semi-abelian categories. As an application, we obtain a categorical proof of the recent characterization of semi-abelian varieties.

Star-regularity and regular completions

“…It is interesting to observe that, unlike normality, protomodularity has the same behaviour with respect to the regular and the exact completions. This is due to the fact that the condition used in [9] for P implies protomodularity of C already when P is merely a projective cover of C. Note, however, that unlike in the present case, the condition used in [9] for P does not imply the same condition on C. Moreover, protomodularity of P does not imply, in general, protomodularity of its regular completion. For instance, the regular completion of the protomodular category of groups is not a protomodular category (while it is still normal by Corollary 4.2).…”

“…Similar questions were studied in [9] with respect to protomodular categories [1] in the place of normal categories. It is interesting to observe that, unlike normality, protomodularity has the same behaviour with respect to the regular and the exact completions.…”

“…Finally, at the end of the paper, we explicitly consider the pointed case, where star-regularity becomes normality. We point out some differences between how the regular completion behaves with respect to normality, and how it behaves with respect to some of the other closely related exactness properties, as established in [21,9,14].…”

Semi-localizations of semi-abelian categories

“…As for binary coproducts, an argument which appeared in [23] can be adapted to our purposes. Any two objects of X ex/reg can be presented as coequalizers of equivalence relations Observe that, in the protomodular context, giving a coequalizer of an equivalence relation is equivalent to giving a cokernel of the corresponding Bourn-normal monomorphism.…”

Goursat completions

“…Having this link in mind, our main interest in studying this subject is to characterize projective covers of certain algebraic categories through simpler properties involving projectives and to relate those properties to the known varietal characterizations in terms of the existence of operations of their varietal theories (when it is the case). Such kind of studies have been done for the projective covers of categories which are: Mal'tsev [11], protomodular and semi-abelian [5], (strongly) unital and subtractive [6].…”