Under forming opposite categories we obtain the notion of an op-fibration fibered in groupoids. In old literature this is sometimes called a “cofibration in groupoids” but that terminology collides badly with the notion of cofibration in homotopy theory and model category theory.
Fibrations in groupoids have a simple characterization in terms of their nerves. Let be the nerve functor and for a morphism in Cat, let be the corresponding morphism in sSet.
The functor is an op-fibration in groupoids precisely if the morphism is a left Kan fibration of simplicial sets, i.e. precisely if for all horn inclusion
\Lambda[n]_i \hookrightarrow \Delta[n]
for all and all smaller than – , we have that every commuting diagram