model structure for (2,1)-sheaves

Model category theory

model category



Universal constructions


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



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

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



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


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

Suppose first that the (2,1)-site CC 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)(2,1)-sheaves.


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

Write [C op,Grpd nat] proj[C^{op}, Grpd_{nat}]_{proj} for the projective model structure on functors on the functor category [C op,Grpd][C^{op}, Grpd].

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

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 WW.

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 CC, for n=2n = 2.


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

Let W={Δ[n]UΔ[n]Un2,UC}W = \{\partial \Delta[n] \cdot U \to \Delta[n] \cdot U| n \geq 2 \in \mathbb{N}, U \in C\} be the set of generating morphisms of weak equivalences on homotopy 1-types.


[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 WW. Then this is a model structure for (2,1)(2,1)-sheaves on CC.

These two model structures are equivalent:



(τN):GrpdNτsSet (\tau \dashv N) : Grpd \stackrel{\overset{\tau}{\leftarrow}}{\underset{N}{\to}} sSet

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

(τ *N *):[C op,Grpd nat] locN *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).


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)(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 (