nLab Malcev category




A Malcev 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).



The category Grp of all groups (including non-abelian groups) is a Malcev category (Borceux & Bourn 2004, Ex. 2.2.4). More generally, for 𝒞\mathcal{C} any category with finite limits, the category Grp(𝒞)Grp(\mathcal{C}) of group objects internal to 𝒞\mathcal{C} is a Malcev category.

More generally, the category of TT-algebras for any Lawvere theory TT which contains a group operation (an Ω\Omega-group). Other examples include the category, HeytHeyt, of Heyting algebras and the category of left closed magmas. The dual category to an elementary topos is a Malcev category. (Borceux & Bourn 2004, Ex. 2.2.5-7) A Malcev variety is a variety of algebras whose category of models is a Malcev category.



In any Malcev category, every internal category is an internal groupoid.


The category of internal categories and functors in a Mal’cev category is Mal’cev category. (Gran 1999, Theorem 3.2)


Last revised on February 23, 2024 at 03:29:20. See the history of this page for a list of all contributions to it.