Spahn Grothendieck construction in HTT (Rev #2)

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

• ‘’right fibration’‘ (HTT, 2..1.) generalizes ‘’fibration in groupoids’‘

• ‘’left fibration’‘ (HTT, 2..1.) generalizes ‘’cofibration in groupoids?’‘

• ‘’cartesian fibration’‘ (HTT, 2.4) generalizes ‘’Grothendieck fibration’‘

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

• contravariant model structure (HTT, 2.1.4) aka. model structure for right fibrations.

• covariant model structure (HTT, 2.1.4) aka. model structure for left fibrations.

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 fibrations?