nLab
global model structure on simplicial presheaves

Idea

For $C$ a category regarded as an (infinity,1)-category, a global model structure on the functor category $[C^{\mathrm{op}},\mathrm{SSet}]$ of simplicial presheaves on $C$ is a presentation for the (infinity,1)-category of (infinity,1)-presheaves on $C$.

The global model structure on simplicial presheaves is a special case of the general notion of global model structure on functors. Its relevance in the general discussion of model structure on simplicial presheaves is mainly its Bousfield localization to a local model structure on simplicial presheaves.

Definition

In every global model structure on simplicial presheaves on $C$ the weak equivalences are objectwise those with respect to the standard model structure on simplicial sets.

So a morphism $f:A\to B$ in $[C^{\mathrm{op}},\mathrm{SSet}]$ is a weak equivalence with respect to a global model structure precisely if for all $U\in \mathrm{Obj}(C)$ the morphism

$$f(U):A(U)\to B(U)$$f(U) : A(U) \to B(U)

is a weak equivalence of simplicial set (i.e. a morphism inducing isomorphisms of simplicial homotopy groups).

There are several choices for how to extend this notion of weak equivalences to an entire model category structure. The two common extreme choices are

• the global projective model structure has as fibrations the objectwise fibrations of the standard model structure on simplicial sets (i.e. the Kan fibrations);

• the global injective model structure has as cofibrations the objectwise cofibrations with respect to the standard model structure on simplicial sets (i.e. the monomorphisms).

The other class of morphisms (cofibrations / fibrations) is in each case fixed by the correspinding lifting property.

Remarks

For $C$ a site, the Bousfield localization of these global model structures at morphisms that induce isomorphisms on all sheaves of simplicial homotopy groups yields the local model structure on simplicial presheaves.

Properties

• Both projective and injective model structures define proper simplicial model categories.

References

See also model structure on simplicial presheaves.

The global projective model structure is originally due to

• A. K. Bousfield and D.M. Kan, Homotopy limits completions and localizations, Springer Lecture Notes in Math. 304 (2nd corrected printing), Springer-Verlag, Berlin–Heidelberg–New York (1987).

The fact that the global injective model structure yields a proper simplicial cofibrantly generated model category is originally due to

• A. Heller, Universal Homotopy Theories (hopf)

The fact that the global projective structure yields a proper simplicial cellular model category is due to Hirschhorn-Bousfield-Kan-Quillen

• P. Hirschhorn, Localizations of Model Categories (web)

A quick review of these facts is on the first few pages of

• Benjamin Blander, Local projective model structure on simplicial presheaves (pdf)