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...

By a classical result of Dominique Bourn, the second cohomology group of a group \( G \) with coefficients in a \( G \)-module \( A \) can be recovered by the fibre over \( A \) of the so-called direction functor \( d=d_0 \), when it is applied to the slice category \( G_p/G \). The higher...