A new characterisation of Goursat categories
type: Item
instance of: scholarly article
title: A new characterisation of Goursat categories
zbMATH Open document ID: 1262.08005
author: Marino Gran
author: Diana Rodelo
published in: Applied Categorical Structures
publication date: 2012-09-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1512.04066
review text: The categories in this paper are finitely complete, regular categories. A regular category \(\mathbb C\) is called a Goursat category if the equivalence relations in \(\mathbb C\) are 3-permutable, i.e., for every pair of equivalence relations \(R\), \(S\) on the same object, the equality \(RSR=SRS\) holds. The crucial result in this paper is that a regular category is a Goursat category if and only if any pushout in this category with vertical split epimorphisms and horizontal regular epimorphisms is a Goursat pushout, i.e., the comparison arrow of the top horizontal morphism in the pushout is a regular epimorphism. This result is used to prove that the validity of the denormalised 3-by-3 lemma is equivalent to the Goursat property (\textit{S. Lack} [Homology Homotopy Appl. 6, No. 1, 1--3 (2004; Zbl 1070.18007)] has shown this in one direction, namely starting with the Goursat category assumption). Finally, the authors give a new categorical proof of the following known result: For a variety \(\mathbb C\) of universal algebras, \(\mathbb C\) is a 3-permutable variety if and only if the algebraic theory \(T\) of \(\mathbb C\) contains two quaternary operations \(p\), \(q\) which satisfy the identities \(p(x,y,y,z)=x\), \(q(x,y,y,z)=z\) and \(p(x,x,y,y)=q(x,x,y,y)\).
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 18C05
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 6084581
zbMATH Keywords: Goursat category
zbMATH Keywords: Goursat pushout
zbMATH Keywords: 3-by-3 lemma
zbMATH Keywords: Mal'tsev condition
MaRDI profile type: Publication
OpenAlex ID: W1996301369
arXiv ID: 1512.04066
cites work: Q4827951
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DOI: 10.1007/S10485-010-9236-X
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description: scientific article; zbMATH DE number 6084581