A new characterisation of Goursat categories
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 \(p(x,x,y,y)=q(x,x,y,y)\).
scientific article; zbMATH DE number 6084581
scientific article