nLab Grothendieck construction for model categories

Idea

The Grothendieck construction may be lifted from categories to model categories, where it serves as a presentation for the $\infty$-Grothendieck construction.

Definition

Let $\mathcal{M}$ be a model category and

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

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

We say that:

Definition

$F$ is relative functor if for any weak equivalence $f$ in $M$, the associated Quillen adjunction $f_! \dashv f^*$ is a Quillen equivalence.

Definition

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

Definition

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

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

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

• an integral equivalence iff

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

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

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

• an integral cofibration iff

  1. $f$ is a cofibration in $\mathcal{M}$

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

• an integral fibration iff

  1. $f$ is a fibration in $\mathcal{M}$

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

Proposition

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

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

Properties

Proposition

Given a proper relative $F \colon \mathcal{M} \to ModelCat$, we can compose with the underlying $(\infty,1)$-functor $Ho \colon ModelCat \to QCat$ with values in (say) quasicategories. Since $F$ is relative, this map takes weak equivalences in $\mathcal{M}$ to equivalences of quasicategories, so it induces a functor of quasicategories $Ho(M) \to Ho(QCat) = (\infty,1)Cat$. The (∞,1)-Grothendieck construction of this functor is then equivalent, over $Ho(M)$, to the underlying $(\infty,1)$-category of the Grothendieck-construction model structure on $\int F$

(Harpaz & Prasma (2015), Proposition 3.1.2)

Examples

Free coproduct completions

Example

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

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

Here $f_!$ forms coproducts of objects in the same fibers of $f$. The corresponding Grothendieck construction is the free coproduct completion of the underlying category $\mathcal{C}$ (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 $f$ in $Set$ being isomorphisms, the associated functors $f_!, f^\ast$ 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 $\mathcal{C}$ is an extensive category, then the objects in this Grothendieck construction may be thought of as bundles of $\mathcal{C}$-objects over sets (under the unique coproduct-preserving embedding $Set \hookrightarrow \mathcal{C}$) and the morphisms as morphisms of such bundles covering possibly non-trivial maps of base sets:

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

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

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

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

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

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

Remark

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 $\mathcal{C}$; analogously for the fibrant objects.

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

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

where

• $\varnothing$ (the empty set) denotes the initial object in $Set$

• $0$ denotes the initial object in the empty-product category, which is the terminal category, so that $0$ is in fact a zero object, indeed it is the only object in that category

  $$\underset{\varnothing}{\prod} \mathcal{C} \;\simeq\; \{0\} \,.$$

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

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

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

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

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

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

This shows that also $0_{\varnothing}$ is the initial object in $\int_S \,\prod_{s \in S}\, \nathcal{C}$. (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 $Set$ are cofibrations while the cofibrations in a product model category are the tuples of cofibrations) that $\mathcal{V}_S$ is cofibant precisely if $\mathcal{V}_s \,\in\, \mathcal{C}$ is so for all $s \in S$.

Example

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

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

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

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 $sSet\text{-}Grpd$ the usual model structure on simplicial groupoids, then the inclusion functor

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

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.

Related pages

• The dual notion is a model structure on sections

• tangent (infinity,1)-topos

• Joyal locus

• parameterized homotopy theory

• parameterized stable homotopy theory

References

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

• Agustí Roig, Model category structures in bifibred categories, Journal of Pure and Applied Algebra 95 2 (1994) 203–223 [doi:10.1016/0022-4049(94)90074-4]

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

• Alexandru E. Stanculescu, Bifibrations and weak factorisation systems, Applied Categorical Structures 20 1 (2012) 19–30 [doi:10.1007/s10485-009-9214-3]

The construction was then generalized in

• Yonatan Harpaz, Matan Prasma, The Grothendieck construction for model categories, Advances in Mathematics 281 (2015) 1306-1363 [arXiv:1404.1852, 10.1016/j.aim.2015.03.031]

and further in

• Pierre Cagne, Paul-André Melliès, On bifibrations of model categories, Advances in Mathematics 370 (2020) 107205 [arXiv:1709.10484, doi:10.1016/j.aim.2020.107205]

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

• Sharon Hollander, A homotopy theory for stacks (arXiv:math.AT/0110247).

Dicussion of further examples:

Integral model structure for graupal Segal-space actions:

• Yonatan Harpaz, Matan Prasma, An integral model structure and truncation theory for coherent group actions, Isr. J. Math. 221 (2017) 511–561 [arXiv:1506.04117, doi:10.1007/s11856-017-1551-6]