nLab Mal'cev category

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 in a Mal’cev category.

In any Mal’cev category, every internal category is a groupoid.

See the monograph Borceux–Bourn.

Revised on August 10, 2013 15:07:09 by Tim Porter (95.147.237.2)