On 2-star-permutability in regular multi-pointed categories

Abstract

2-star-permutable categories were introduced in a joint work with Z. Janelidze and A. Ursini as a common generalisation of regular Mal'tsev categories and of normal subtractive categories. In the present article we first characterise these categories in terms of what we call star-regular pushouts. We then show that the 3 × 3 Lemma characterising normal subtractive categories and the Cuboid Lemma characterising regular Mal'tsev categories are special instances of a more general homological lemma for star-exact sequences. We prove that 2-star-permutability is equivalent to the validity of this lemma for a star-regular category.

Beck–Chevalley condition and Goursat categories

“…By applying Proposition 4.1 in [15] we conclude that the squares (B) are pullbacks. By the usually called Barr-Kock theorem [1] we conclude that (A) is a pullback, as desired.…”

Projective Covers of 2-star-permutable Categories

“…In order to prove our main result, we will need to impose an additional condition on the regular category E regarding the behavior of regular epimorphisms with respect to N -kernels. This condition is familiar from the literature (see [8,10,13]) and is indeed mild enough that it includes all examples of interest. We now proceed to introduce the necessary notions.…”

“…Then, in a more general multi-pointed context, the pertinent notion becomes that of a star-exact sequence, which unifies the pointed and nonpointed versions, and allows for the aforementioned multi-pointed homological lemmas. Finally, in [13] the notion of 2-star-permutable category was studied as a common extension of both regular subtractive and regular Mal'tsev categories and characterizations of these categories via diagrams such as regular pushouts were generalized to a multi-pointed context. In the present note we want to add to this list a characterization of projective covers of regular 2-star-permutable multi-pointed categories.…”

“…In this note we look at regular Mal'tsev and regular subtractive categories as special cases of the notion of 2-star-permutable category, following the line of research in [10], [13]. The aim here is to obtain a characterization of the projective covers of 2-star-permutable multi-pointed categories, thus unifying and subsuming the known characterizations in the Mal'tsev [18] and subtractive [12] settings.…”