# Spahn Grothendieck construction in HTT

The Grothendieck construction relates

• categories of fibrations with contravariant functor categories with codomain $Cat$

• categories of fibrations in groupoids with contravariant functor categories with codomain $Grpd$

and dually

• categories of cofibrations with covariant functor categories with codomain $Cat$

• categories of cofibrations in groupoids with covariant functor categories with codomain $Grpd$

To categorify this constructions to $(\infty,1)$-category theory we have the following dictionary of notions

On the level of model categories this is accomplished in the following way:

## Requisites

### enriched model categories

($sSet$ is an excellent model category Example A.3.2.18)

Proposition 2.1.4.8 (p.65): For a simplicial set $S$ the ovecategory $sSet/S$ is a simplicial enriched model category. A.3.1.7

### (co)fibrations

cartesian fibration?

Last revised on June 29, 2012 at 11:55:28. See the history of this page for a list of all contributions to it.