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

Let $C$ be a simplicial model category

From the axioms of enriched model category it follows that for $X\in C$ cofibrant and $A\in C$ fibrant, the simplicial set $C(X,A)$ given by the $\mathrm{sSet}$-enrichement is fibrant in the standard model structure on simplicial sets, hence is a Kan complex. In fact, 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 a $\mathrm{sSet}$-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 week 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 charaxcterization 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

Proof

This is prop 1.3

• 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)

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 $A$.

Proof

This is theorm 3.1 in

• 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)

Warning on terminology

The term simplicial model category for the notion described here is entirely standard, but in itself a bit subobtimal. 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

• Hirschhorn, Model categories and their localization

section A.3 in

• Jacob Lurie, Higher Topos Theory