nLab
simplicial model category

Idea

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 $C$ together with the structure of a model category on its underlying category $C_0$ 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 $C^{\circ }\hookrightarrow C$ 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 $C_0$.

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

Definition

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

This means that a simplicial model category is

• an sSet-enriched category

  • which is powered and copowered over sSet

• with the structure of a model category on the underlying category $C_0$

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

• and such that this fibration is an acyclic fibration whenever either $i$ or $p$ are acyclic.

Properties

Enrichment, tensoring, and cotensoring

Let $𝒞$ 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 condition that each make $𝒞$ into a simplicial model category are equivalent

1. the tensoring $\otimes :𝒞\times \mathrm{sSet}\to 𝒞$ is a left Quillen bifunctor;

2. for any cofibration $X\to Y$ and fibration $A\to B$ in $𝒞$, the induced morphism

   $$𝒞(Y,A)\to 𝒞(X,A){\times }_{𝒞(X,B)}𝒞(Y,B)$$\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 $X\to Y$ in $\mathrm{sSet}$ and fibration $A\to B$ in $𝒞$, the induced morphism

   $$A^Y\to A^X{\times }_{B^X}{B}^Y$$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 $𝒞$ now be a simplicial model category.

Corollary

If $A\in 𝒞$ is fibrant, and $X\hookrightarrow Y$ is a cofibration in sSet, then

$$X^Y\to X^A$$X^Y \to X^A

is a fibration in $𝒞$.

Proof

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

$$A^Y\to A^X{\times }_{{*}^X}{*}^Y$$A^Y \to A^X \times_{{*}^X} {*}^Y

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

Similarly we have

Corollary

If $X\in 𝒞$ is cofibrant and $A\in 𝒞$ is fibrant, then $𝒞(Y,X)$ is fibrant in sSet, hence is a Kan complex.

Proof

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

$$𝒞(X,A)\to 𝒞(\varnothing ,A){\times }_{𝒞(\varnothing ,*)}𝒞(X,*)$$\mathcal{C}(X, A) \to \mathcal{C}(\emptyset, A) \times_{\mathcal{C}(\emptyset,*)} \mathcal{C}(X,*)

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

$$𝒞(X,A)\to *.$$\mathcal{C}(X,A) \to * \,.

Remark

For $X$ and $A$ any two objects and $QX$ and $PA$ a cofibrant and fibrant replacement, respectively, $𝒞(QX,PA)$ is the correct derived hom-space between $X$ and $A$. In particular the full $\mathrm{sSet}$-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 $C$.

Examples

The category ${\mathrm{sSet}}_{\mathrm{Quillen}}$ is a monoidal model category and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.

For $C$ any small sSet-enriched category and $A$ simplicial combinatorial model category, the global model structure on functors $[C^{\mathrm{op}},A{]}_{\mathrm{proj}}$ and $[C^{\mathrm{op}},A{]}_{\mathrm{inj}}$ are themselved simplicial combinatorial model categories.

The left Bousfield localization of these at any set of morphisms is again a combinatorial simplicial model category. Large clases of examples arise this way. In particular for $A={\mathrm{sSet}}_{\mathrm{Quillen}}$ itself this yields the model structure on simplicial presheaves.

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.

Simplicial Quillen equivalent models

While some model categories do not admit an ${\mathrm{sSet}}_{\mathrm{Quillen}}$-enrichment, for large classes of model categories one can find an Quillen equivalence to a model cateory that does have an ${\mathrm{sSet}}_{\mathrm{Quillen}}$-enrichment.

Theorem

Every left proper combinatorial model category is Quillen equivalent to a simplicial model category.

More precisely: let $A$ be a

• left proper;

• combinatorial or cellular

model category. Then the category $\mathrm{sA}:=[{\Delta }^{\mathrm{op}},A]$ of simplicial objects in $A$ carries the structure of a simplicial model category with respect to its canonical $\mathrm{sSet}$-enrichment, such that the functor ${\mathrm{ev}}_0:\mathrm{sA}\to A$ that sends a simplicial object to its degree 0 piece exhibits a Quillen equivalence

$$c_{*}:A\overrightarrow{\leftarrow }:{\mathrm{ev}}_0.$$c_* : A \stackrel{\to}{\leftarrow} : ev_0 \,.

In $\mathrm{sA}$ we have

• the weak equivalences are those morphisms that induce weak equivalences in $A$ under homotopy colimit over $\Delta$;

• the cofibrations are cofibrations in the Reedy model structure on $[{\Delta }^{\mathrm{op}},A]$.

• the fibrant objects are precisely the Reedy-fibrant objects all whose face and degenarcy maps are weak equivalences in $A$.

Proof

This is theorem 1.2 of

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

The simplicial enrichment is theorem 6.1

The characterization of the fibrant objects is theorem 5.7.

Theorem

There is a similar model structure on $\mathrm{sA}$ where instead the cofibrations are the cofibrations with respect to the projective global model structure on functors on $[{\Delta }^{\mathrm{op}},A]$.

In this the fibrant objects are precisely the simplicial objects that are degreewise fibrant in $A$, and for which all face and degeneracy maps are weak equivalences in $A$.

The identity functor established a Quillen equivalence between this model structure and the one discussed above

$$s{A}_{\mathrm{Reedy},\mathrm{loc}}\overleftarrow{\to }{\mathrm{sA}}_{\mathrm{proj},\mathrm{loc}}.$$s A_{Reedy, loc} \stackrel{\leftarrow}{\to} sA_{proj, loc} \,.

Proof

This is theorem 1.3 in the above article.

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 (RezkSchwedeShipley,prop 1.3).

Theorem

(uniqueness)

Let $A$ be a model category. Then there is at most one model category structure on $sA=[{\Delta }^{\mathrm{op}},A]$ such that

• every morphism that is degreewise a weak equivalence in $A$ 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 $A$.

This is (RezkSchwedeShipley, theorm 3.1).

Warning on terminology

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 different (though not much) from a simplicial object in Cat.

The other caveat is that there are different model category structures on sSet and hence even the term $\mathrm{sSet}$-enriched model category is ambiguous.

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

References

section 9.1.5 of

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

• Charles Rezk, Stefan Schwede, Brooke Shipley, Simplicial structures on model categories and functors American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575 (jstor)

section A.3 in

• Jacob Lurie, Higher Topos Theory