Algebraic exponentiation and action representability for V -groups

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.