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fibration fibered in groupoids in nLab

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Contents

Idea

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

Definition

Definition

A fibration fibered in groupoids is a functor p:E→Bp : E \to B such that the corresponding (strict) functor B op→B^{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:Cat→sSetN : Cat \to sSet be the nerve functor and for p:E→Bp : 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:E→Bp : 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] i↪Δ[n] \Lambda[n]_i \hookrightarrow \Delta[n]

for all n∈ℕn \in \mathbb{N} and all ii smaller than nn (0≤i<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) } \,.

References

Review:

Last revised on November 9, 2024 at 14:48:54. See the history of this page for a list of all contributions to it.