A fibration fibered in groupoids is a functor$p : E \to B$ such that the corresponding (strict) functor $B^{op} \to$Cat classifying $p$ under the Grothendieck construction factors through the inclusion Grpd$\hookrightarrow$Cat.

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.

Properties

Fibrations in groupoids have a simple characterization in terms of their nerves. Let $N : Cat \to sSet$ be the nerve functor and for $p : E \to B$ a morphism in Cat, let $N(p) : N(E) \to N(B)$ be the corresponding morphism in sSet.

Then

Proposition

The functor $p : E \to B$ is an op-fibration in groupoids precisely if the morphism $N(p) : N(E) \to N(B)$ is a left Kan fibration of simplicial sets, i.e. precisely if for all horn inclusion

$\Lambda[n]_i \hookrightarrow \Delta[n]$

for all $n \in \mathbb{N}$ and all $i$smaller than $n$ – $0 \leq i \lt n$, we have that every commuting diagram