Relative Mal'tsev categories (Q2859012)

In [J. Algebra 371, 132--155 (2012; Zbl 1275.18030], the first, the second and the fourth author of this paper introduced a framework containing a condition called relative Mal'tsev axiom, which is a must in the axiomatic study of higher-dimensional extensions in [the first author et al., Adv. Math. 217, No. 5, 2231--2267 (2008; Zbl 1140.18012); J. Algebra 324, No. 8, 1771--1789 (2010; Zbl 1225.18012)] and their relationship to simplicial resolutions. On the other hand, the third author of this paper, in collaboration with others, has been extending the framework of relative homological algebra to non-additive categories [\textit{T. Janelidze}, J. Homotopy Relat. Struct. 1, No. 1, 185--194 (2006; Zbl 1127.18007); Appl. Categ. Struct. 17, No. 4, 373--386 (2009; Zbl 1180.18005); \textit{M. Zawadowski}, J. Pure Appl. Algebra 216, No. 8--9, 1932--1942 (2012; Zbl 1269.18003)]. It was shown in [\textit{A. Carboni} et al., Appl. Categ. Struct. 1, No. 4, 385--421 (1993; Zbl 0799.18002)] that in every regular category \(\mathcal{A}\), the condition of every simplicial object being Kan is tantamount to that of the category \(\mathcal{A}\) being a Mal'tsev category. This leads naturally to this paper, which investigates the relative Mal'tsev axiom in the context of regular categories. It is shown particularly that the relative Mal'tsev axiom is equivalent to the condition of every \(\mathcal{E}\)-simplicial object abiding by a relative Kan property.