A class of exactness properties characterized via left Kan extensions
In [J. Algebra 583, 38--88 (2021; Zbl 1472.18002)], the author and \textit{Z. Janelidze} studied exactness properties on a regular category [\textit{M. Barr} et al., Exact categories and categories of sheaves. Cham: Springer (1971; Zbl 0223.18009)]. The properties of being a Mal'tsev category [\textit{A. Carboni} et al., J. Pure Appl. Algebra 69, No. 3, 271--284 (1990; Zbl 0722.18005)], or more generally, an \(n\)-permutable category [\textit{A. Carboni} et al., Appl. Categ. Struct. 1, No. 4, 385--421 (1993; Zbl 0799.18002)] are examples of such categories. The author extracted the Mal'tsev conditions characterizing varieties of universal algebras abiding by them, and generalized the theory of approximate co-operations from [Zbl 1172.18001; Zbl 1188.18002; Zbl 1401.18013; Zbl 1423.18012]. The principal objective in this paper is to generalize these results in several aspects. A synopsis of the paper consisting of seven sections goes as follows. \S 1 describes the exactness properties under investigation in the paper, recalling some aspects of the theory of exactness sequents [the author and \textit{Z. Janelidze}, Adv. Math. 377, Article ID 107484, 56 p. (2021; Zbl 1452.18005)]. \S 2 constructs, for each such exactness property, the universal \(\mathcal{B}_{n} \)-structure associated to each object of a finitely bicomplete category. \S 3 deals with the characterization of finitely bicomplete categories abiding by such an exactness property by means of this universal \(\mathcal{B}_{n} \)-structure associated to each object, giving rise to the generalization of the theory of approximate co-operations in the regular context. \S 4 extends this characterization in the varietal case, giving an easy way to find the algebraic condition associated to each of these exactness properties. \S 5 is devoted to some examples, namely, protomodularity, involution-rigidness [\textit{Z. Janelidze} and \textit{N. Martins-Ferreira}, J. Algebra Appl. 16, No. 4, Article ID 1750074, 14 p. (2017; Zbl 1376.08002)] and the Mal'tsev property. \S 6 treats the pointed context, giving additional examples. \S 7 makes further remarks, among which a generalization of the characterization in the varietal case to the case of essentially algebraic categories [\textit{J. Adámek} et al., Cah. Topologie Géom. Différ. Catégoriques 29, No. 3, 175--192 (1988; Zbl 0659.18010)] is given.
essentially algebraic category Mal'tsev condition
scientific article; zbMATH DE number 7389894
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