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.

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.

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

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

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

For instance HTT, prop. 2.1.1.3.

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]

Last revised on May 7, 2023 at 07:04:22. See the history of this page for a list of all contributions to it.