nLab model structure for (2,1)-sheaves

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

Then let finally

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

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

Write

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

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

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

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

$$(\tau_* \dashv N_*) : [C^{op}, Grpd_{nat}]_{loc} \underoverset{\underset{N_*}{\to}}{\overset{\tau_*}{\leftarrow}}{\simeq} [C^{op}, sSet_{Quillen}]_{loc, W}$$

that is a Quillen equivalence.

This appears as (Hollander, theorem 5.4).

Related concepts

• model structure for $(2,1)$-sheaves

• model structures for $(\infty,1)$-sheaves

  • model structure on simplicial presheaves

  • model structure on simplicial sheaves

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.

• Sharon Hollander, A homotopy theory for stacks (arXiv:math.AT/0110247)

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)