nLab fibration fibered in groupoids

Idea

A Grothendieck fibration fibered in groupoids – usually called a category fibered in groupoids – is a Grothendieck fibration $p : E \to B$ all whose fibers are groupoids.

Definition

Definition

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

$$\array{ \Lambda[n]_i &\to& N(E) \\ \downarrow && \downarrow^{\mathrlap{N(p)}} \\ \Delta[n] &\to& N(B) }$$

has a lift

$$\array{ \Lambda[n]_i &\to& N(E) \\ \downarrow &\nearrow& \downarrow^{\mathrlap{N(p)}} \\ \Delta[n] &\to& N(B) } \,.$$

Proof

For instance HTT, prop. 2.1.1.3.

Related concepts

• Grothendieck fibration, Grothendieck construction

• discrete fibration

• left fibration, right fibration, Cartesian fibration, Kan fibration

References

Review:

• Angelo Vistoli, §3.3 in: Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry – Grothendieck's FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, math.AG/0412512]