A Mal’cev category is a left exact category (= having finite limits) in which any reflexive internal relation is an equivalence relation. Equivalently, the fibers of its fibration of points are unital (equivalently the fibers of the fibration of points are strongly unital).
Examples include the category of groups, and in fact the category of $T$-algebras for any theory $T$ which contains a group operation (an $\Omega$-group). Other examples include the category, $Heyt$, of Heyting algebras and the category of left closed magmas. The dual category to an elementary topos is a Mal’cev category. A Malʹcev variety is a variety of algebras whose category of models is a Mal’cev category.
In any Mal’cev category, every internal category is a groupoid.
Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004 (monograph)
Dominique Bourn, From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories, Compact Textbooks in Mathematics, Birkhäuser 2017 (textbook)
Last revised on August 4, 2019 at 15:32:34. See the history of this page for a list of all contributions to it.