Algebraic exponentiation and action representability for V-groups
Maria Manuel Clementino, Andrea Montoli
TL;DR
This paper studies the category of $V$-groups for a cartesian quantale $V$, showing it is $S$-protomodular with respect to product-structure points, action representable with representing object $V\text{-}{\rm Aut}(X)$, and locally algebraically cartesian closed (S-LACC). The approach uses enrichment over $V$-Cat, interpreting split extensions as $V$-functors into $V\text{-}{\rm Aut}(X)$ and employing right Kan extensions to construct adjoints to change-of-base functors. The main contributions are the relative action representability and LACC results for $V$-groups, generalizing known results for Schreier points in Mon and preordered groups to a setting of cartesian quantales. The findings provide a unified categorical framework for non-abelian homological properties in enriched group-like structures, with preordered groups as a key instance. These results illuminate how product-structure points govern representations of actions and the algebraic closure properties in the enriched setting.
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.