Grothendieck construction for model categories


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



The Grothendieck construction may be lifted from categories to model categories, where it serves as a presentation for the (infinity,1)-Grothendieck construction.


Let MM be a model category and F:MModelCatF:M\to ModelCat a pseudofunctor, where ModelCatModelCat is the 2-category of model categories, Quillen adjunctions pointing in the direction of their left adjoints, and mate-pairs of natural isomorphisms. Assume furthermore that:

On the Grothendieck construction F\int F we define a morphism (f,ϕ):(A,X)(B,Y)(f,\phi):(A,X) \to (B,Y), where f:ABf:A\to B in MM and ϕ:f !(X)Y\phi:f_!(X) \to Y in F(B)F(B), to be:

Then these classes of maps make F\int F a model category.


Given a proper relative F:MModelCatF:M\to ModelCat, we can compose with the underlying (,1)(\infty,1)-category functor Ho:ModelCatQCatHo:ModelCat \to QCat with values in (say) quasicategories. Since FF is relative, this map takes weak equivalences in MM to equivalences of quasicategories, so it induces a functor of quasicategories Ho(M)Ho(QCat)=(,1)CatHo(M) \to Ho(QCat) = (\infty,1)Cat. The (∞,1)-Grothendieck construction of this functor is then equivalent, over Ho(M)Ho(M), to the underlying (,1)(\infty,1)-category of the Grothendieck-construction model structure on F\int F; this is Harpaz-Prasma, Proposition 3.1.2.


The first model category version of the Grothendieck construction was given in

This article (Roig 94) had a mistake, which was fixed in

The construction was then generalized in

Another approach is found in

For the special case of pseudofunctors with values in groupoids, a model category version of the Grothendieck construction was discussed in