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model structure for (2,1)-sheaves

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Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

(∞,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A model structure of (2,1)-sheaves is a model category presentation of the (2,1)-category of (2,1)-sheaves over some site or (2,1)-site.

Definition

There are several equivalent ways to set up a model category structure for (2,1)-sheaves.

Suppose first that the (2,1)-site C is just a 1-category, hence just a site.

The following definition first defines a model presentation for (2,1)-presheaves (1-truncated (∞,1)-presheaves) and then localizes at the covering morphisms in order to obtain the (2,1)-sheaves.

Definition

Write Grpd for the category of small category groupoids and functors between them. Write Grpd nat for the natural model structure on groupoids.

Write [C op,Grpd nat] proj for the projective model structure on functors on the functor category [C op,Grpd].

Let W={C({U i})→j(U)} be the set of Cech nerve projections in [C op,Grpd] for each covering family {U i→U} in the site C.

Then let finally

[C op,Grpd nat] proj,loc[C^{op}, Grpd_{nat}]_{proj,loc}

be the left Bousfield localization at the set of morphisms W.

The following definition first gives the presentations for (∞,1)-sheaves and then further restricts the 1-truncated objects in there, preseting the (n,1)-topos inside the full (∞,1)-topos over C, for n=2.

Definition

Write [C op,sSet Quillen] loc for a local model structure on simplicial presheaves on C, the one which presents the (∞,1)-category of (∞,1)-sheaves on C.

Let W={∂Δ[n]⋅U→Δ[n]⋅U∣n≥2∈ℕ,U∈C} be the set of generating morphisms of weak equivalences on homotopy 1-types.

Write

[C op,sSet Quillen] loc,W[C^{op}, sSet_{Quillen}]_{loc,W}

for the left Bousfield localization of the model structure for (∞,1)-sheaves at the morphisms W. Then this is a model structure for (2,1)-sheaves on C.

These two model structures are equivalent:

Proposition

Let

(τ⊣N):Grpd→N←τsSet(\tau \dashv N) : Grpd \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}} sSet

be the nerve functor and its left adjoint τ. Postcomposition with this induces a Quillen adjunction

(τ *⊣N *):[C op,Grpd nat] loc≃→N *←τ *[C op,sSet Quillen] loc,W(\tau_* \dashv N_*) : [C^{op}, Grpd_{nat}]_{loc} \underoverset{\underset{N_*}{\to}}{\overset{\tau_*}{\leftarrow}}{\simeq} [C^{op}, sSet_{Quillen}]_{loc, W}

that is a Quillen equivalence.

This appears as (Hollander, theorem 5.4).

References

A model structure on presheaves of groupoids Quillen equivalent to the left Bousfield localization of the local model structure for (∞,1)-sheaves at morphisms that are weak equivalences of homtopy 1-types is in.

A discussion of (2,1)-sheaves/stacks as 1-truncated objects in the full model structure for (∞,1)-sheaves is in

  • J. F. Jardine, Stacks and the homotopy theory of simplicial sheaves , Homology Homotopy Appl. Volume 3, Number 2 (2001), 361-384. (project euclid)

Revised on January 23, 2011 21:29:57 by Urs Schreiber (89.204.137.66)
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