nLab
fibration fibered in groupoids

Contents

Idea

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

Definition

Definition

A fibration fibered in groupoids is a functor p:EBp : E \to B such that the corresponding (strict) functor B opB^{op} \to Cat classifying pp 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:CatsSetN : Cat \to sSet be the nerve functor and for p:EBp : E \to B a morphism in Cat, let N(p):N(E)N(B)N(p) : N(E) \to N(B) be the corresponding morphism in sSet.

Then

Proposition

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

Λ[n] iΔ[n] \Lambda[n]_i \hookrightarrow \Delta[n]

for all nn \in \mathbb{N} and all ii smaller than nn0i<n0 \leq i \lt n, we have that every commuting diagram

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

has a lift

Λ[n] i N(E) N(p) Δ[n] N(B). \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.

Revised on December 11, 2011 05:38:47 by Urs Schreiber (212.87.29.231)