Algebraic exponentiation and action representability for V-groups

TL;DR

This paper investigates the categorical properties of V-groups, showing they are locally algebraically cartesian closed and that their actions are representable, extending the understanding of algebraic structures in enriched category theory.

Contribution

It demonstrates that the category of V-groups is locally algebraically cartesian closed and that actions are representable, generalizing previous results to categories enriched over a cartesian quantale.

Findings

V-groups form a locally algebraically cartesian closed category.

Actions in V-groups are representable.

The results apply to preordered groups as a special case.

Abstract

We show that the category of V-groups, where V is a cartesian quantale, so in particular the category of preordered groups, is locally algebraically cartesian closed with respect to the class of points underlying the product V-category structure. We obtain this by observing that such points correspond to (V-Cat)-enriched functors from a V-group, seen as a one-object V-category, to the category V-Grp of V-groups. Moreover, we show that the actions corresponding to points underlying the product V-category structure are representable.