nLab
Malcev category

Contents

Contents

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. 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.

(Borceux & Bourn 2004, Ex. 2.2.6)

More generally, the category of TT-algebras for any 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. 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.

References

Last revised on September 22, 2021 at 05:14:13. See the history of this page for a list of all contributions to it.