nLab
Grothendieck construction for model categories

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

Idea

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

Definition

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:

  • FF is relative, meaning that whenever f:ABf:A\to B is a weak equivalence in MM, then the Quillen adjunction f !:F(A)F(B):f *f_! : F(A) \rightleftarrows F(B) : f^* is a Quillen equivalence. (That is, FF is a relative functor?.)

  • FF is proper, meaning that whenever f:ABf:A\to B is an acyclic cofibration (resp. an acyclic fibration) in MM, then f !f_! (resp. f *f^*) preserves weak equivalences.

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:

  • a weak equivalence if f:ABf:A\to B is a weak equivalence in MM and f !(QX)f !(X)ϕYf_!(Q X) \to f_!(X) \xrightarrow{\phi} Y is a weak equivalence in F(B)F(B), where QQ is a cofibrant replacement. Since f !f *f_!\dashv f^* is a Quillen equivalencen by relativeness, this is equivalent to the dual condition that Xϕ^f *(Y)f *(RY)X \xrightarrow{\hat{\phi}} f^*(Y) \to f^*(R Y) is a weak equivalence in F(A)F(A).

  • a cofibration if ff is a cofibration in MM and ϕ:f !(X)Y\phi : f_!(X)\to Y is a cofibration in F(B)F(B).

  • a fibration if ff is a fibration in MM and the adjunct ϕ^:Xf *(Y)\hat{\phi} : X\to f^*(Y) is a fibration in F(A)F(A).

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

Properties

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.

References

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

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

  • Alexandru E. Stanculescu, Bifibrations and weak factorisation systems, Applied Categorical Structures 20.1, p.19–30, (2012) (doi:10.1007/s10485-009-9214-3)

The construction was then generalized in

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

Last revised on February 15, 2019 at 07:07:31. See the history of this page for a list of all contributions to it.