nLab Malcev category

Definition

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).

Examples

Example

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

More generally, the category of $T$-algebras for any Lawvere 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 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.

Properties

Proposition

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

Proposition

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

Related notions

• subtractive category

• Goursat category

• internal crossed module

References

• Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004 (doi:10.1007/978-1-4020-1962-3)

• Dominique Bourn, From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories, Compact Textbooks in Mathematics, Birkhäuser 2017 (textbook)

• Marino Gran, Internal categories in Mal’cev categories, Journal of Pure and Applied Algebra 143 1-3 (1999) 221-229 [doi:10.1016/S0022-4049(98)00112-1]