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model structure on simplicial sheaves

Idea

For $S$ a small site let $\mathrm{SimpSh}(S)$ be the full category of simplicial presheaves on $S$ which satisfy the sheaf condition: the category of simplicial sheaves.

The local model category structure on $\mathrm{SimpSh}(S)$ is originally due to Joyal. It is closely related to the local model structure on simplicial presheaves.

Local injective model structure

Theorem

There is a proper closed simplicially enriched model category structure on $\mathrm{SimpSh}(S)$ such that

• cofibrations are precisely the objectwise cofibrations (monomorphisms) of simplicial sets;

• weak equivalences are the local weak equivalences of the underlying simplicial presheaves as defined at model structure on simplicial presheaves.

See theorem 5 in Jardine07.

Theorem

The inclusion of sheaves into simplicial presheaves $\mathrm{SimpSh}(S)\hookrightarrow \mathrm{SimpPr}(S)$ and the sheafification functor $\mathrm{SimpPr}(S)\to \mathrm{SimpSh}(S)$ constitute a Quillen equivalence with respect to the above first local model structure on $\mathrm{SimpPr}(S)$ ans the local model structure on simplicial sheaves.

See Jardine07, theorem 5.

References

• Sjoerd Crans, Quillen closed model structure for sheaves, J. Pure Appl. Algebra 101 (1995), 35-57 (web)

see also model structure on simplicial presheaves for more literature

Jardine’s lectures

• Jardine07 J. Jardine, Field Lectures: Simplicial presheaves (pdf)

discuss the Quillen equivalence between the model structure on simplicial sheaves and the model structure on simplicial presheaves.