# nLab global model structure on simplicial presheaves

model category

## Definitions

• category with weak equivalences

• weak factorization system

• homotopy

• small object argument

• resolution

• ## Universal constructions

• homotopy Kan extension

• Bousfield-Kan map

• ## Refinements

• monoidal model category

• enriched model category

• simplicial model category

• cofibrantly generated model category

• algebraic model category

• compactly generated model category

• proper model category

• stable model category

• ## Producing new model structures

• on functor categories (global)

• on overcategories

• Bousfield localization

• transferred model structure

• Grothendieck construction for model categories

• ## Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

• ## Model structures

• Cisinski model structure
• ### for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

• Strom model structure?
• Thomason model structure

• model structure on presheaves over a test category

• model structure on simplicial groupoids

• on cubical sets

• related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

• ### for $n$-groupoids

• for 1-groupoids

• ### for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

• ### for $\infty$-algebras

#### general

• on monoids

• on algebas over a monad

• on modules over an algebra over an operad

• #### specific

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

• model structure on dg-modules

• ### for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on presheaves of spectra

• ### for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

• ### for stable $(\infty,1)$-categories

• on dg-categories
• ### for $(\infty,1)$-operads

• on modules over an algebra over an operad

• ### for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

• ### for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

• model structure for (2,1)-sheaves/for stacks

• # Contenta

## 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

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

## References

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

Last revised on September 20, 2013 at 22:57:19. See the history of this page for a list of all contributions to it.