nLab
global model structure on simplicial presheaves

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contenta

Idea

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

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

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

In particular, if CC carries the structure of a site, then

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

Definition

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

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

f(U):A(U)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 other class of morphisms (cofibrations / fibrations) is in each case fixed by the corresponding lifting property.

Properties

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 Heller Homotopy 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

Revised on September 20, 2013 22:57:19 by Tim Porter (95.147.236.5)