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 $B^{\mathrm{op}}\to$ 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 $N:\mathrm{Cat}\to \mathrm{sSet}$ 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

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

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

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

has a lift

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

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