A class of exactness properties characterized via left Kan extensions
type: Item
instance of: scholarly article
title: A class of exactness properties characterized via left Kan extensions
zbMATH Open document ID: 1471.18010
published in: Journal of Pure and Applied Algebra
publication date: 2021-09-02T00:00:00Z
review text: 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. (2…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18E13
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 18C30
Mathematics Subject Classification ID: 08A55
zbMATH DE Number: 7389894
zbMATH Keywords: exactness property Kan extension
zbMATH Keywords: strong epimorphism
zbMATH Keywords: approximate operation
zbMATH Keywords: essentially algebraic category Mal'tsev condition
author: Pierre-Alain Jacqmin
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106784
OpenAlex ID: W3174872303
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DOI: 10.1016/J.JPAA.2021.106784
description: scientific article; zbMATH DE number 7389894