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 * ''[[nLab:right fibration]]'' ([[HTT, 2.]].1.) generalizes ''[[nLab:fibration in groupoids]]'' * ''[[nLab:left fibration]]'' ([[HTT, 2.]].1.) generalizes ''[[nLab:cofibration in groupoids]]'' * ''[[nLab:cartesian fibration]]'' ([[HTT, 2.]]4) generalizes ''[[nLab:Grothendieck fibration]]'' On the level of model categories this is accomplished in the following way: * contravariant model structure ([[HTT, 2.]]1.4) aka. [[nLab:model structure for right fibrations]]. * covariant model structure ([[HTT, 2.]]1.4) aka. [[nLab: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 fibration]]