A Grothendieck fibration fibered in groupoids – usually called a category fibered in groupoids – is a Grothendieck fibration all whose fibers are groupoids.
A fibration fibered in groupoids is a functor such that the corresponding (strict) functor Cat classifying under the Grothendieck construction factors through the inclusion Grpd 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 be the nerve functor and for a morphism in Cat, let be the corresponding morphism in sSet.
Then
The functor is an op-fibration in groupoids precisely if the morphism is a left Kan fibration of simplicial sets, i.e. precisely if for all horn inclusion
for all and all smaller than – , we have that every commuting diagram
has a lift
For instance HTT, prop. 2.1.1.3.
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Last revised on May 7, 2023 at 07:04:22. See the history of this page for a list of all contributions to it.