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simplicial model category

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Enriched category theory

Contents

Idea

The term simplicial model category is short for sSet Quillen{}_{Quillen}-enriched model category.

A simplicial model category is a model or presentation for an (∞,1)-category that is half way in between a bare model category and a Kan complex-enriched category.

Specifically, a simplicial model category is an sSet-enriched category CC together with the structure of a model category on its underlying category C 0C_0 such that both structures are compatible in a reasonable way.

One important use of simplicial model categories comes from the fact that the full sSet-subcategory C CC^\circ \hookrightarrow C on the fibrant-cofibrant objects – which is not just sSet-enriched but actually Kan complex-enriched – is the (∞,1)-category-enhancement of the homotopy category of the model category C 0C_0.

For generalizations of this construction with sSet replaced by another monoidal model category see enriched homotopical category.

Remark

The term simplicial model category for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of simplicially enriched category, which is a proper special case of a simplicial object in Cat (that for which the simplicial set of objects is discrete).

The other caveat is that there are different model category structures on sSet and hence even the term sSetsSet-enriched model category is ambiguous.

For instance the model structure for quasi-categories is an sSetsSet-enriched model category, but not for the standard Quillen model structure on the enriching category: since sSet JoyalsSet_{Joyal} is a closed monoidal model category it is enriched over itself, hence is a sSet JoyalsSet_{Joyal}-enriched model category, not an sSet QuillensSet_{Quillen}-enriched one. So in the standard terminology, sSet JoyalsSet_{Joyal} is not a “simplicial model category”.

Definition

Definition

A simplicial model category is an enriched model category which is enriched over sSet QuillensSet_{Quillen}: the category sSet equipped with its standard model structure on simplicial sets.

Spelled out, this means that a simplicial model category is

  • an sSet-enriched category

  • with the structure of a model category on the underlying category C 0C_0

  • such that for every cofibration i:ABi : A \to B and every fibration p:XYp : X \to Y in C 0C_0 the morphism of simplicial sets C(B,X)i *×p *C(A,X)× C(A,Y)C(B,Y)C(B,X) \stackrel{i^* \times p_*}{\to} C(A,X) \times_{C(A,Y)} C(B,Y) is a fibration;

    • and such that this fibration is an acyclic fibration whenever either ii or pp are acyclic.

Properties

Enrichment, tensoring, and cotensoring

Let 𝒞\mathcal{C} be a category equipped with the structure of a model category and with that of an sSet-enriched category with is tensored and cotensored over sSet.

Proposition

The following conditions – that each make 𝒞\mathcal{C} into a simplicial model category – are equivalent:

  1. the tensoring :𝒞×sSet𝒞\otimes : \mathcal{C} \times sSet \to \mathcal{C} is a left Quillen bifunctor;

  2. for any cofibration XYX \to Y and fibration ABA \to B in 𝒞\mathcal{C}, the induced morphism

    𝒞(Y,A)𝒞(X,A)× 𝒞(X,B)𝒞(Y,B) \mathcal{C}(Y, A) \to \mathcal{C}(X, A) \times_{\mathcal{C}(X,B)} \mathcal{C}(Y,B)

    is a fibration, and is in addition a weak equivalence if either of the two morphisms is;

  3. for any cofibration XYX \to Y in sSetsSet and fibration ABA \to B in 𝒞\mathcal{C}, the induced morphism

    A YA X× B XB Y A^Y \to A^X \times_{B^X} B^Y

    is a fibration, and is in addition a weak equivalence if either of the two morphisms is.

This follows directly from the defining properties of tensoring and cotensoring.

We list in the following some implications of these equivalent conditions.

Let 𝒞\mathcal{C} now be a simplicial model category.

Corollary

If A𝒞A \in \mathcal{C} is fibrant, and XYX \hookrightarrow Y is a cofibration in sSet, then

A YA X A^{Y} \to A^{X}

is a fibration in 𝒞\mathcal{C}.

Proof

Apply prop. 1 to the case of the cofibration XYX \to Y and the fibration A*A \to *, where “**” denotes the terminal object. This yields that

A YA X× * X* Y A^Y \to A^X \times_{{*}^X} {*}^Y

is a fibration. But * Y=* X=*{*}^Y = {*}^X = {*} and hence the claim follows.

Similarly we have

Corollary

If X𝒞X \in \mathcal{C} is cofibrant and A𝒞A \in \mathcal{C} is fibrant, then 𝒞(Y,X)\mathcal{C}(Y,X) is fibrant in sSet, hence is a Kan complex.

Proof

Apply prop. 1 to the cofibration X\emptyset \to X, where “\emptyset” denotes the initial object, and to the fibration A*A \to * to find that

𝒞(X,A)𝒞(,A)× 𝒞(,*)𝒞(X,*) \mathcal{C}(X, A) \to \mathcal{C}(\emptyset, A) \times_{\mathcal{C}(\emptyset,*)} \mathcal{C}(X,*)

is a fibration. But since \emptyset is initial and ** is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration

𝒞(X,A)*. \mathcal{C}(X,A) \to * \,.

Derived hom-spaces

Proposition

For XX and AA any two objects and QXQ X and PAP A a cofibrant and fibrant replacement, respectively, 𝒞(QX,PA)\mathcal{C}(Q X, P A) is the correct derived hom-space between XX and AA (see the discussion there). In particular the full sSetsSet-enriched subcategory on cofibrant fibrant objects is therefore an sSet-enriched category which is fibrant in the model structure on simplicially enriched categories. Its homotopy coherent nerve is a quasi-category. All this are intrinsic incarnatons of the (∞,1)-category that is presented by CC.

Examples

General

Simplicial Quillen equivalent models

While many model categories do not admit an sSet QuillensSet_{Quillen}-enrichment, for large classes of model categories one can find a Quillen equivalence to a model category that does have an sSet QuillensSet_{Quillen}-enrichment.

These are constructed as Bousfield localization of Reedy model structures on the category of simplicial objects in the given model category.

Definition

Let CC be a

By the discussion at cofibrantly generated model category in the section Presentation and generation there exists a small set EObj(C)E \subset Obj(C) of objects that detect weak equivalences. For some such choice of EE, let

S:={e(Δ[k]Δ[l])} eE,([k][l])ΔMor([Δ op,sSet]) S := \{ \; e \cdot (\Delta[k] \to \Delta[l]) \; \}_{e \in E, ([k] \to [l]) \in \Delta} \subset Mor([\Delta^{op}, sSet])

where eΔ[k]:[n] Δ([n],[k])ee \cdot \Delta[k] : [n] \mapsto \coprod_{\Delta([n],[k])} e.

Write

[Δ op,C] proj,S[Δ op,C] proj [\Delta^{op}, C]_{proj, S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj}

for the left Bousfield localization of the projective model structure on functors at this set SS of morphisms.

Similarly, write

[Δ op,C] Reedy,S[Δ op,C] Reedy [\Delta^{op}, C]_{Reedy, S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{Reedy}

for the left Bousfield localization of the Reedy model structure at SS.

Lemma

Let CC be a cofibrantly generated model category.

If X[Δ op,C]X \in [\Delta^{op}, C] is degreewise cofibrant and has all structure maps being weak equivalences, then all X ihocolimXX_i \to hocolim X are weak equivalences.

Hence XconsthocolimXX \to const\,hocolim X is a weak equivalence.

This appears as (Dugger, prop. 5.4 corollary 5.5).

Theorem

The model structures from def. 2 have the following properties.

  1. The weak equivalences in both are precisely those morphisms which become weak equivalences under homotopy colimit over Δ op\Delta^{op}.

  2. The fibrant objects in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the face and degeneracy maps are weak equivalences in CC.

  3. The colimit/constant adjoint functors

    (lim const):Cconstlim [Δ op,C] proj,S (\lim_{\to} \dashv const) : C \stackrel{\overset{\lim_\to}{\leftarrow}}{\underset{const}{\to}} [\Delta^{op}, C]_{proj, S}

    constitute a Quillen equivalence, the identity functors constitute a Quillen equivalence

    [Δ op,C] Reedy,Sidid[Δ op,C] proj,S, [\Delta^{op}, C]_{Reedy, S} \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} [\Delta^{op}, C]_{proj, S} \,,

    and the constant/limit adjoint functors (since Δ op\Delta^{op} has an initial object the limit is evaluation in degree 0) constitute a Quillen equivalence

    (constev 0):[Δ op,C] Reedy,Sev 0constC; (const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C \,;
  4. The canonical sSet-enrichment/tensoring/powering of the category of simplicial objects [Δ op,C][\Delta^{op}, C] makes [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} (but not in general [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S}) into a simplicial model category.

This is (Dugger, theorem 5.2, theorem 5.7, theorem 6.1).

So in particular every left proper combinatorial model category is Quillen equivalent to a simplicial model category.

Proof

We first show that the fibrant objects in [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S} are the objectwise fibrant objects all whose structure maps are weak equivalences in CC. The argument for the fibrant objects in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} is directly analogous.

By general properties of left Bousfield localization, the fibrant objects in [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S} are the projective fibrant objects XX for which all induced morphisms on derived hom spaces

Hom [Δ op,C] proj(s(Δ[k]Δ[l]),X) \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}(s \cdot(\Delta[k] \to \Delta[l]), X)

are weak equivalences. Since ss is cofibrant in CC by definition, also sΔ[k]s \cdot \Delta[k] is cofibrant in [Δ op,C] proj[\Delta^{op}, C]_{proj}.

So for X[Δ op,C] projX \in [\Delta^{op}, C]_{proj} fibrant, let X [Δ op,[Δ op,C]]X_\bullet \in [\Delta^{op}, [\Delta^{op}, C]] be a simplicial framing for it. Notice that this means that for all nn \in \mathbb{N} also X ([n])X_\bullet([n]) is a simplicial framing for X([n])X([n]). This is because

  1. constXX const X \to X_\bullet being a weak equivalence means that for all nn the morphism XX nX \to X_n is a weak equivalence, which means that for all kk the morphism X([k])X n([k])X([k]) \to X_n([k]) is a weak equivalence.

  2. X X_\bullet being fibrant in [Δ op,[Δ op,C] proj] Reedy[\Delta^{op}, [\Delta^{op}, C]_{proj}]_{Reedy} means that for all nn\in \mathbb{N} the morphism X Δ[n]X Δ[n]X_{\Delta[n]} \to X_{\partial \Delta[n]} is a fibration in [Δ op,C] proj[\Delta^{op}, C]_{proj}, hence that for all kk \in \mathbb{N} the morphism X Δ[n]([k])X Δ[n]([k])X_{\Delta[n]}([k]) \to X_{\partial \Delta[n]}([k]) is a fibration in CC, hence that X([k])X([k]) is Reedy fibrant.

Then we find

Hom [Δ op,C] proj(s(Δ[k]Δ[l]),X) Hom [Δ op,C](s(Δ[k]Δ[l]),X ) Hom C(s,X ([l])X ([k])) Hom C(s,X([l])X([k])). \begin{aligned} \mathbb{R}Hom_{[\Delta^{op}, C]_{proj}}( s \cdot(\Delta[k] \to \Delta[l]), X) & \simeq Hom_{[\Delta^{op}, C]}(s \cdot(\Delta[k] \to \Delta[l]), X_\bullet) \\ & \simeq Hom_{C}(s , X_\bullet([l]) \to X_\bullet([k])) \\ & \simeq \mathbb{R}Hom_C(s, X([l]) \to X([k])) \,. \end{aligned}

By assumption on the set SS, this implies the claim.

\,

Now we show that the weak equivalences in [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S} are precisely those morphisms that become weak equivalences under the homotopy colimit.

By functorial cofibrant resolution and two-out-of-three, it is sufficient to show that this holds for morphisms between cofibrant objects.

By lemma 1, we have weak equivalences

Hom [Δ op,C](A,X)Hom [Δ op,C](A,constZ)Hom C(lim A,Z) \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) \stackrel{\simeq}{\to} \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) \stackrel{\simeq}{\to} \mathbb{R}Hom_{C}(\lim_\to A , Z)

seen by computing the derived homs by simplicial framings.

Now, by properties of left Bousfield localization, ABA \to B is a weak equivalence if for all SS-local objects XX the morphism Hom(AB,X)\mathbb{R}Hom(A \to B, X) is a weak equivalence. Looking at the diagram

Hom [Δ op,C](A,X) Hom [Δ op,C](A,constZ) Hom C(lim A,Z) Hom [Δ op,C](B,X) Hom [Δ op,C](B,constZ) Hom C(lim B,Z) \array{ \mathbb{R}Hom_{[\Delta^{op}, C]}(A,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(A, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to A , Z) \\ \uparrow && \uparrow && \uparrow \\ \mathbb{R}Hom_{[\Delta^{op}, C]}(B,X) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{[\Delta^{op}, C]}(B, const Z) &\stackrel{\simeq}{\to}& \mathbb{R}Hom_{C}(\lim_\to B , Z) }

we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant ZCZ \in C, which is the case if lim Alim B\lim_\to A \to \lim_\to B is a weak equivalence. Since AA and BB here are cofibrant in [Δ op,C] proj[\Delta^{op}, C]_{proj}, the colimits here are indeed homotopy colimits (as discussed there).

\,

Now we discuss that (lim const):C[Δ op,C] proj,S(\lim_\to \dashv const): C \to [\Delta^{op}, C]_{proj,S} is a Quillen equivalence. First observe that on the global model structure const:C[Δ op,C] projconst : C \to [\Delta^{op}, C]_{proj} is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the derived functor of the left adjoint lim \lim_\to, namely the homotopy colimit, takes the localizing set SS to weak equivalences in CC. Therefore the assumptions of the discussion at Quillen equivalence - Behaviour under localization are met, and hence it follows that (lim const)(\lim_\to \dashv const) descends as a Quillen adjunction also to the localization.

To see that this is a Quillen equivalence, it is sufficient to show that for A[Δ op,C] projA \in [\Delta^{op}, C]_{proj} cofibrant and ZCZ \in C fibrant, a morphism lim AZ\lim_\to A \to Z is a weak equivalence in CC precisely if the adjunct AconstZA \to const Z becomes a weak equivalence under the homotopy colimit.

For this notice that we have a commuting diagram

hocolimA hocolim(constZ) colimA colim(constZ) Z \array{ hocolim A &\to& hocolim(const Z) \\ \downarrow && \downarrow \\ colim A &\to& colim (const Z) & \simeq Z }

and so our statement follows (by 2-out-of-3) once we know that the vertical morphisms here are weak equivalences. The left one is because AA is cofibrant, by assumption, as before. To see that the right one is, too, consider the factorization

Z(constZ) ihocolim(constZ)Z Z \to (const Z)_i \to hocolim (const Z) \to Z

of the identity on ZZ, for any ii \in \mathbb{N}. By lemma 1 the first morphism is a weak equivalence, and hence so is the morphism in question.

\,

Now we show that the weak equivalences in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} are the hocolim-equivalences.

By a general result on functoriality of localization, we have that the (idid):[Δ op,C] Reedy,S[Δ op,C] proj,S(id \dashv id ) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj,S} is at least a Quillen adjunction.

Let then ABA \to B be a morphism in [Δ op,C][\Delta^{op}, C] and consider two fibrant replacements

A A¯ A^ B B¯ B^, \array{ A &\to& \bar A &\to & \hat A \\ \downarrow && \downarrow && \downarrow \\ B &\to& \bar B &\to& \hat B } \,,

where the first one (A¯B¯\bar A \to \bar B) is taken in [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S} and the second (\hat A \to \hat B) in [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}.

Assume first that ABA \to B is a hocolim-equivalence. Then so is A^B^\hat A \to \hat B, because the horizontal morphisms are all objectwise weak equivalences. But A^\hat A and B^\hat B are fibrant in [Δ op,C] Reedy[\Delta^{op}, C]_{Reedy}, hence in [Δ op,C] proj[\Delta^{op}, C]_{proj} by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in [Δ op,C] proj,S[\Delta^{op}, C]_{proj,S}. By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure – so A^B^\hat A \to \hat B is indeed even an objectwise weak equivalence. It follows then that so is A¯B¯\bar A \to \bar B, which is therefore in partiular a weak equivalence in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy, S}. Finally the left horizontal morphisms are also weak equivalences in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S}, by the above Quillen adjunction. So finally by 2-out-of-3 in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} it follows that also ABA \to B is a weak equivalence there.

By an analogous diagram chase, one shows the converse implication holds, that ABA \to B being a weak equivalence in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} implies that it is a hocolim-equivalence.

With this now it is clear that the identity adjunction above is in fact a Quillen equivalence.

\,

Finally we show that (constev 0):[Δ op,C] Reedy,Sev 0constC(const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C is a Quillen equivalence.

First, it is immediate to check that const:C[Δ op,C] Reedyconst : C \to [\Delta^{op}, C]_{Reedy} is left Quillen, and since id:[Δ op,C] Reedy[Δ op,C] Reedy,Sid : [\Delta^{op}, C]_{Reedy} \to [\Delta^{op}, C]_{Reedy,S} is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction.

To see that it is a Quillen equivalence, let ACA \in C be cofibrant and X[Δ op,C] Reedy,SX \in [\Delta^{op}, C]_{Reedy,S} be fibrant – which by the above means that it is a simplicial resolution – and consider a morphism constAXconst A \to X. We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if AX 0A \to X_0 is a weak equivalence in CC.

To that end, find a cofibrant resolution constA˜X˜const \tilde A \to \tilde X of constAXconst A \to X in [Δ op,C] proj[\Delta^{op}, C]_{proj} and consider the diagram

A A˜ colim(constA˜) X 0 X˜ 0 colimX˜. \array{ A &\stackrel{\simeq}{\leftarrow}& \tilde A &\stackrel{\simeq}{\to}& colim(const \tilde A) \\ \downarrow && \downarrow && \downarrow \\ X_0 &\stackrel{\simeq}{\leftarrow}& \tilde X_0 &\stackrel{\simeq}{\to}& colim \tilde X } \,.

The colimits on the right compute the homotopy colimit. By 2-out-of-3 it follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is.

,\,,

Finally it remains to show that [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} is a simplicially enriched model category. (…)

Remark

(uniqueness)

Let CC be a model category. Then there is a unique model category structure on sC=[Δ op,C]s C = [\Delta^{op}, C] such that

  • every morphism that is degreewise a weak equivalence in CC is a weak equivalence;

  • the cofibrations are those of the Reedy model structure;

  • the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in CC.

This is (Rezk-Schwede-Shipley, theorem 3.1).

Proof

By theorem 1 at least one such model structure exists. By the discussion at model category – Redundancy of the axioms, the classes of cofibrations and fibrant objects already determine a model category structure.

Corollary

For CC any left proper combinatorial model category, the derived hom-space between two objects X,AX, A may be computed by

  • choosing a cofibrant replacement X^\hat X of XX in CC;

  • choosing a Reedy fibrant replacement A^\hat A of constAconst A in [Δ op,C][\Delta^{op}, C] such that all face and degeneracy maps are weak equivalences,

setting

Maps(X,A):[n]Hom C(X^,A^ n). Maps(X,A) : [n] \mapsto Hom_C(\hat X, \hat A_n) \,.
Proof

By theorem 1 we may compute the derived hom space in [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} after the inclusion const:C[Δ op,C]const : C \to [\Delta^{op}, C]. Since by that theorem [Δ op,C] Reedy,S[\Delta^{op}, C]_{Reedy,S} is a simplicial model category, by prop. 2 the derived hom space is given by the simplicial function complex between a cofibrant replacement of constXconst X and a fibrant replacement of constAconst A. If X^\hat X is cofibrant, then constX^const \hat X is already Reedy cofibrant, and by the theorem A^\hat A as stated is a a fibrant resolution of constAconst A. Finally, the theorem says that the simplicial function complex is given by

[Δ op,C](constX^,A^) n =Hom [Δ op,C]((constX^)Δ[n],A^) Hom [Δ op,C]((constX^),A^ Δ[n]) Hom C(X^,A^ n). \begin{aligned} [\Delta^{op}, C](const \hat X, \hat A)_n & = Hom_{[\Delta^{op}, C]}((const \hat X) \cdot \Delta[n], \hat A) \\ & \simeq Hom_{[\Delta^{op}, C]}((const \hat X) , \hat A^{\Delta[n]}) \\ & \simeq Hom_C(\hat X, \hat A_n) \end{aligned} \,.

There is also a version for stable model categories:

Theorem

Every proper cofibrantly generated stable model category is Quillen equivalent to a simplicial model category

This is (Rezk-Schwede-Shipley, prop 1.3).

Combinatorial simplicial model categories

A particularly important type of simplicial model categories are those that are also combinatorial model categories.

A combinatorial simplicial model category is precisely a presentation for a locally presentable (∞,1)-category. See there for more details.

References

A standard textbook reference is section 9.1.5 of

  • P. Hirschhorn, Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs , American Mathematical Society, 2009.

Original results are in

  • Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. vol. 353, number 12 (2001), 5003-5027. (pdf)

section A.3 in

Revised on July 19, 2013 02:26:20 by Urs Schreiber (89.204.138.192)