On the categorical behaviour of $V$-groups

We consider compatible group structures on a $V$-category, where $V$ is a quantale, and we study the topological and algebraic properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when $V$ is a frame, symmetric $V$-groups satisfy very strong categorical-algebraic properties, typical of the category of groups. In particular, symmetric $V$-groups form a protomodular category.