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 SSet.

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.

Examples

• The most well-developed model/presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves on an ordinary (i.e. 1-categorical) site $S$ is the Kan complex-enriched category $C^{\circ }\hookrightarrow C$ for $C=[S^{\mathrm{op}},\mathrm{SSet}]$ the simplicial model category of simplicial presheaves on $S$ equippped with the model structure on simplicial presheaves.

References

section 9.1.5 of

• Hirschhorn, Model categories and their localization

section A.3 in

• Jacob Lurie, Higher Topos Theory