On the ``Smith is Huq'' condition in \(S\)-protomodular categories
The authors investigate the ``Smith is Huq'' condition in the context of \(S\)-protomodular categories. These are defined as follows. A point in a pointed finitely complete category is a pair \((f,s)\) of morphisms such that \(fs=1\). The class of points determines a fibration (the functor associating to each point its codomain). Taking a class \(S\) of points stable under pull-backs determines a subfibration. A category \(\mathbb C\) is \(S\)-protomodular if any object in the full subcategory of points of \(\mathbb C\) whose objects are in \(S\) is a strongly split epimorphism, and the set of such points is closed under finite limits. In particular, the category of monoids with operations is \(S\)-protomodular for \(S\) the class of Schreier split epimorphisms. The ``Smith is Huq'' condition is: any two equivalence relations centralise each other in the Smith-Pedicchio sense [\textit{M. C. Pedicchio}, J. Algebra 177, No. 3, 647--657 (1995; Zbl 0843.08004)] if and only if their normalisations commute in the \textit{S. A. Huq} sense [Q. J. Math., Oxf. II. Ser. 19, 363--389 (1968; Zbl 0165.03301)]. The authors study the Smith is Huq condition in the \(S\)-relative context and compare it with others, concerning properties of internal reflexive graphs and of the kernel-functor. Some of these fail in the \(S\)-relative context. The authors also prove that the \(S\)-relative Smith is Huq condition holds if \(S\) is the class of Schreier split epimorphisms in any category of monoids with operations. Finally, they describe some consequences of the Smith is Huq condition in terms of internal categories and groupoids.
scientific article; zbMATH DE number 6692920
scientific article