Algebraic exponentiation and action representability for V-groups

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.

💡 Research Summary

The paper investigates the categorical properties of V‑groups, where V is a cartesian quantale (i.e., a complete lattice equipped with a tensor product that is the cartesian meet ∧ and unit ⊤). The authors aim to show that the category V‑Grp of V‑groups is locally algebraically cartesian closed (LACC) and action‑representable, but only with respect to a carefully chosen class S of points, namely those whose middle object carries the product V‑category structure.

The exposition begins with a review of protomodular categories and the notion of a point (a split epimorphism together with a chosen splitting). The fibration of points Pt(–) is introduced, and protomodularity is characterized by the conservativity of all change‑of‑base functors. Since many interesting algebraic categories fail to be protomodular in full generality, the authors recall the relative notion of S‑protomodularity: a class S of points stable under pullback gives rise to a sub‑fibration S‑cod, and a category is S‑protomodular when the points in S are strong and SPt(–) is closed under finite limits. This framework captures, for example, Schreier points in the category of monoids and product points in the category of pre‑ordered groups (OrdGrp).

Next, the paper defines V‑categories and V‑groups. A V‑relation a : X × X → V satisfies reflexivity (κ ≤ a(x,x)) and transitivity (a(x,x′)⊗a(x′,x″) ≤ a(x,x″)). A V‑group is a group (X,+,0,i) equipped with a V‑relation a such that the group operation + : (X,a)⊗(X,a) → (X,a) is a V‑functor. Morphisms are V‑functors that are also group homomorphisms, forming the category V‑Grp. Important examples include: V = 2 (the two‑element lattice) giving OrdGrp (pre‑ordered groups); V = (