New wide classes of weakly Mal'tsev categories
The paper continues the study of weakly Mal'tsev categories, introduced by the author in [Theory Appl. Categ. 21, 91--117 (2008; Zbl 1166.18005)] as an extension of Mal'tsev categories (see, e.g., \textit{F. Borceux} and \textit{D. Bourn} [Mal'cev, protomodular, homological and semi-abelian categories. Mathematics and its Applications (Dordrecht) 566. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1061.18001)]). In particular, the author shows certain classes of categories to be weakly Mal'tsev. More precisely, he introduces the class of weakly Mal'tsev algebras with cancellation as a framework, which encompasses all the known examples of algebraic structures making weak Mal'tsev categories (Proposition 1 on page 744). Starting with the observation that the dual of the category of topological spaces is weakly Mal'tsev, the author arrives at his main result that the dual of a category with pullbacks and pushouts of split monomorphisms along split monomorphisms, in which pullbacks of local coproducts are jointly epimorphic, is weakly Mal'tsev (Theorem 1 on page 745). As a consequence, he concludes that the dual of any quasi-adhesive category is weakly Mal'tsev (Proposition 2 on page 748). The paper ends with the results that, first, the dual of an extensive category with pullbacks, pushouts of split monomorphisms along split monomorphisms, and pullback stable epimorphisms, is weakly Mal'tsev (Corollary 1 on page 749), which includes, in particular, the category of topological spaces; and, second, the dual of a solid quasi-topos (one with disjoint coproducts) is a weakly Mal'tsev category (Corollary 2 on page 750). The paper is well written, provides some of its required preliminaries, and will be of use to all those researchers, who are interested in categorical algebra.
scientific article; zbMATH DE number 6502200
scientific article