nLab global model structure on simplicial presheaves

Idea

The global model structure on simplicial presheaves $[C^{op}, sSet_{Quillen}]$ on a small category $C$ is the global model structure on functors on $C$ with values in $sSet_{Quillen}$, the standard model structure on simplicial sets.

It presented the (∞,1)-category of (∞,1)-functors from $C^{op}$ to ∞Grpd, hence the (∞,1)-category of (∞,1)-presheaves on $C$.

The left Bousfield localizations of $[C^{op}, sSet]_{proj}$ are, up to Quillen equivalence, precisely the combinatorial model categories.

In particular, if $C$ carries the structure of a site, then

• the left Bousfield localization of $[C^{op}, sSet_{Quillen}]$ at Cech covers is the Cech model structure on simplicial presheaves;

• the left Bousfield localization at hypercovers is the local model structure on simplicial presheaves.

These localizations present the topological localization and hypercompletion of the (∞,1)-topos of $(\infty,1)$-presheaves on $C$ to the corresponding (∞,1)-topos of (∞,1)-sheaves/∞-stacks on $C$.

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.

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

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

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 corresponding lifting property.

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

• Alex HellerHomotopy Theories, no. 383, Memoirs Amer. Math. Soc., Amer. Math. Soc., 1988.

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)

Details on the projective global model structure is in

• Daniel Dugger, Universal Homotopy Theories