Some remarks on Maltsev and Goursat categories
type: Item
instance of: scholarly article
title: Some remarks on Maltsev and Goursat categories
zbMATH Open document ID: 0799.18002
DOI: 10.1007/BF00872942
author: Max Kelly
author: A. Carboni
author: M. C. Pedicchio
published in: Applied Categorical Structures
publication date: 1994-06-05T00:00:00Z
review text: Let \({\mathcal A}\) be a regular category, i.e., a finitely complete category with a regular epimorphic-monomorphic factorization stable under pulling back. A relation on an object \(A\) of \({\mathcal A}\) is a subobject \(R\) of \(A \times A\). Two such relations \(R\) and \(S\) can be composed to get a relation \(RS\) on \(A\). If \(R\) and \(S\) are equivalence relations on \(A\), we get two increasing sequences of relations on \(A\): \[ \Delta_ A \leq R \leq RS \leq RSR \leq RSRS \leq \dots \] \[ \Delta_ A \leq S \leq SR \leq SRS \leq SRSR \leq \dots \] whose terms are generally distinct. However it may happen that these sequences are stationary, for any relations \(R\) and \(S\) on any object \(A\) of \({\mathcal A}\). In this case, the category is said to be \(n\)-permutable for some adequate \(n \in \mathbb{N}\). For \(n = 2\), i.e., \(RS = SR\), the category is called a Maltsev category. For \(n = 3\), i.e., \(RSR = SRS\), the category is called a Goursat category. Numerous and various characterizations and examples of these categories are given.
Mathematics Subject Classification ID: 18B10
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 554359
zbMATH Keywords: congruence
zbMATH Keywords: variety
zbMATH Keywords: Kan complex
zbMATH Keywords: regular category
zbMATH Keywords: equivalence relations
zbMATH Keywords: Maltsev category
zbMATH Keywords: Goursat category
reviewed by: Yves Diers
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00872942
OpenAlex ID: W2050717037
description: scientific article; zbMATH DE number 554359