We will consider compatible group structures on a V-category, where V is a quantale, and we will explore some categorical 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 cartesian quantale, symmetric V-groups satisfy very strong categorical-algebraic properties, typical of the category of groups, while the whole category of V-groups satisfies similar properties relatively to a suitable class of split epimorphisms, similarly to what happens for the category of monoids.

Joint work with Maria Manuel Clementino.