nLab 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

general \infty-algebras

specific \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 \infty -Grothendieck construction.


Let \mathcal{M} be a model category and

(1) ModCat X F(X) f f ! f * Y F(Y) \array{ \mathcal{M} &\longrightarrow& Mod Cat \\ X &\mapsto& F(X) \\ \Big\downarrow\mathrlap{{}^{f}} && \mathllap{^{f_!}}\Big\downarrow {}^{\dashv} \Big\uparrow\mathrlap{{}^{f^\ast}} \\ Y &\mapsto& F(Y) }

a pseudofunctor, where ModelCat ModelCat is the 2-category of model categories, Quillen adjunctions (pointing in the direction of their left adjoints f !f_!), and conjugate transformations of adjoints (mate-pairs of natural isomorphisms).

We say that:


FF is relative functor if for any weak equivalence ff in MM, the associated Quillen adjunction f !f *f_! \dashv f^* is a Quillen equivalence.


FF is “proper”, if when f:ABf \colon A\to B is an acyclic cofibration (resp. an acyclic fibration) in \mathcal{M} then f !f_! (resp. f *f^*) preserves all weak equivalences.


(integral model structure)
Given a pseudofunctor as in (1), we say that a morphism

(f,ϕ):(X,A)(Y,B) (f,\phi) \;\colon\; (X,A) \longrightarrow (Y,B)

in its Grothendieck construction XF(X)\int_{X \in \mathcal{M}} F(X) (where f:XYf \colon X\to Y in \mathcal{M} and ϕ:f !(A)B\phi \colon f_!(A) \to B in F(Y)F(Y)) is:

  • an integral equivalence iff

    1. f:XYf \colon X \to Y is a weak equivalence in \mathcal{M}

    2. f !(QA)f !(A)ϕBf_!(Q A) \to f_!(A) \xrightarrow{\phi} B is a weak equivalence in F(Y)F(Y), where Q()Q(-) denotes cofibrant replacement.

      (Since f !f *f_!\dashv f^* is a Quillen equivalence by relativeness, this is equivalent to the adjunct condition that Aϕ˜f *(B)f *(RB)A \xrightarrow{\widetilde{\phi}} f^*(B) \to f^*(R B) is a weak equivalence in F(X)F(X) for fibrant replacement R()R(-).)

  • an integral cofibration iff

    1. ff is a cofibration in \mathcal{M}

    2. ϕ:f !(A)B\phi \colon f_!(A) \to B is a cofibration in F(Y)F(Y).

  • an integral fibration iff

    1. ff is a fibration in \mathcal{M}

    2. the adjunct ϕ˜:Af *(B)\widetilde{\phi} \colon A \to f^*(B) is a fibration in F(X)F(X).


If the pseudofunctor (1) is relative (Def. ) and proper (Def. ) then the classes of maps in Def. make XF(X)\int_{X \in \mathcal{M}} F(X) a model category.

This is Harpaz & Prasma (2015), Theorem 3.0.12.



Given a proper relative F:ModelCatF \colon \mathcal{M} \to ModelCat, we can compose with the underlying ( , 1 ) (\infty,1) -functor Ho:ModelCatQCatHo \colon ModelCat \to QCat with values in (say) quasicategories. Since FF is relative, this map takes weak equivalences in \mathcal{M} 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

(Harpaz & Prasma (2015), Proposition 3.1.2)


Free coproduct completions


(model structure on indexed sets of objects)
For 𝒞\mathcal{C} any model category, consider the pseudofunctor on Set which assigns categories of indexed sets of objects of 𝒞\mathcal{C}, equipped with the product model structure:

Set ModCat S sS𝒞 f f ! f * T sS𝒞 \array{ Set &\longrightarrow& ModCat \\ S &\mapsto& \underset{s \in S}{\prod} \mathcal{C} \\ \Big\downarrow{{}^\mathrlap{f}} && \mathllap{{}^{f_!}}\Big\downarrow {}^{\dashv} \Big\uparrow\mathrlap{{}^{f^*}} \\ T &\mapsto& \underset{s \in S}{\prod} \mathcal{C} }

Here f !f_! forms coproducts of objects in the same fibers of ff. The corresponding Grothendieck construction is the free coproduct completion of the underlying category 𝒞\mathcal{C} (see there).

If we regard Set as equipped with its trivial model structure (whose weak equivalences are the isomorphisms and all morphisms are both fibrations as well as cofibrations) then this is evidently a relative and proper functor in the sense of Def. , Def. (since with weak equivalences ff in SetSet being isomorphisms, the associated functors f !,f *f_!, f^\ast are certainly Quillen equivalences but in fact are plain equivalences of categories compatible with the model structure and hence also preserve all weak equivalences).

Therefore with Prop. the integral model structure on the Grothendieck construction exists. If 𝒞\mathcal{C} is an extensive category, then the objects in this Grothendieck construction may be thought of as bundles of 𝒞\mathcal{C}-objects over sets (under the unique coproduct-preserving embedding Set𝒞Set \hookrightarrow \mathcal{C}) and the morphisms as morphisms of such bundles covering possibly non-trivial maps of base sets:

(2) SSetsS𝒞={sSX s ϕ tTY t S f T} \int_{S \in Set} \underset{s \in S}{\prod} \mathcal{C} \;\;\;\; = \;\;\;\; \left\{ \begin{array}{ccc} \underset{s \in S}{\coprod} X_s &\overset{\phi}{\longrightarrow}& \underset{t \in T}{\coprod} Y_t \\ \big\downarrow && \big\downarrow \\ S &\underset{f}{\longrightarrow}& T \end{array} \right\}

Unwinding the definition of the integral model structure in this case, gives that such a morphism ϕ f\phi_f in this Grothendieck construction is:

  • a fibration iff all the components X sY f(s)X_s \longrightarrow Y_{f(s)} are fibrations in 𝒞\mathcal{C} for all sSs \in S,

  • a cofibration iff all the co-components sf 1({t})X sY t\underset{s \in f^{-1}(\{t\})}{\coprod} X_s \longrightarrow Y_t are cofibrations in 𝒞\mathcal{C}, for all tTt \in T,

  • a weak equivalence iff ff is a bijection of index sets and all the (co)components maps — which in this case are all of the form X sX f(s)X_s \longrightarrow X_{f(s)} — are weak equivalences in 𝒞\mathcal{C}

    (here the composition with (co)fibrant replacements can be omitted, since, as above, f !,f *f_!, f^\ast preserve such resolution weak equivalences).


The cofibrant objects in the integral model structure on a free coproduct completion (Exp. ) are exactly the tuples of cofibrant objects in the coefficient model structure 𝒞\mathcal{C}; analogously for the fibrant objects.

To see this, observe that the initial object in the Grothendieck construction of Exp. is

0 (Set,0𝒞), 0_{\varnothing} \;\coloneqq\; \Big( \varnothing \,\in\, Set ,\; 0 \,\in\, \underset{\varnothing}{\prod} \mathcal{C} \Big) \,,


Now given any object 𝒱 S SSetsS𝒞\mathscr{V}_S \,\in\, \int_{S \in Set}\;\underset{s \in S}{\prod} \, \mathcal{C} observe that there is a unique morphism

0 𝒱 S. 0_{\varnothing} \longrightarrow \mathscr{V}_S \,.

Namely, unwinding the definitions, such a morphism is a dependent pair consisting of

  1. a morphism f:Sf \,\colon\,\varnothing \longrightarrow S in Set,

  2. a morphism ϕ˜:f !0𝒱\widetilde{\phi} \,\colon\, f_! 0 \longrightarrow \mathscr{V} in sS𝒞\underset{s \in S}{\prod} \, \mathcal{C}

But both of these exist uniquely, by the fact that their respective domains are initial objects, using here that f !f_!, being a left adjoint, preserves colimits and hence initial objects

This shows that also 0 0_{\varnothing} is the initial object in S sSnathcalC\int_S \,\prod_{s \in S}\, \nathcal{C}. (Of course this also follows by the general formula for colimits in a Grothendieck construction specialized to the empty colimit.) And it shows (since all morphisms in the trivial model structure on SetSet are cofibrations while the cofibrations in a product model category are the tuples of cofibrations) that 𝒱 S\mathcal{V}_S is cofibant precisely if 𝒱 s𝒞\mathcal{V}_s \,\in\, \mathcal{C} is so for all sSs \in S.


(model structure on skeletal simplicial groupoids) As a special case of Exp. , consider 𝒞sGrpd\mathcal{C} \coloneqq sGrpd the model structure on simplicial groups.

Notice that forming simplicial delooping groupoids is a fully faithful functor from sGrp to the 1-category of sSet-enriched groupoids (Dwyer-Kan’s“simplicial groupoids”)

sGrp sSet-Grpd 𝒢 B𝒢 \array{ sGrp &\longrightarrow& sSet\text{-}Grpd \\ \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G} }

which extends to identify the Grothendieck construction (2) in the present case with the full subcategory of disjoint unions of simplicial delooping groupoids, hence with that of simplicial skeletal groupoids:

(3) SsSsGrp sSet-Grpd skl sSet-Grpd (𝒢 s) sS sSB𝒢 s \array{ \int_S \underset{s \in S}{\prod} sGrp &\;\simeq\;& sSet\text{-}Grpd_{skl} &\xhookrightarrow{\phantom{--}}& sSet\text{-}Grpd \\ \big(\mathcal{G}_s\big)_{s \in S} &\mapsto& \underset{s \in S}{\coprod} \mathbf{B}\mathcal{G}_s }

Since (assuming the axiom of choice in the underlying Sets) every sSet-enriched groupoid is DK-equivalent to a skeletal simplicial groupoid, the full subcategory inclusion (3) is also a Dwyer-Kan equivalence of sSet-enriched categories.

Moreover, if we consider on sSet-GrpdsSet\text{-}Grpd the usual model structure on simplicial groupoids, then the inclusion functor

sSet-Grpd sklsSet-Grpd sSet\text{-}Grpd_{skl} \hookrightarrow sSet\text{-}Grpd

preserves weak equivalences and fibrations. However, since there is no left adjoint to this functor (due to the choices involved in passing to a skeleton such an adjoint exists only as a pseudofunctor, hence on the level of 2-category theory) this is not a right Quillen functor, and in particular not a Quillen equivalence.


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

and further in

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

Dicussion of further examples:

Integral model structure for graupal Segal-space actions:

Last revised on November 2, 2023 at 08:22:26. See the history of this page for a list of all contributions to it.