nLab
simplicial sheaf

Definition

A simplicial sheaf $A$ is equivalently

• a simplicial object

  $$A\in \mathrm{SSh}(C):=[{\Delta }^{\mathrm{op}},\mathrm{Sh}(C)]$$A \in SSh(C) := [\Delta^{op}, Sh(C)]

  in a category of sheaves $\mathrm{Sh}(C)$ for some site $C$;

• a simplicial presheaf

  $$A\in \mathrm{SPSh}(C):=[{\Delta }^{\mathrm{op}},\mathrm{PSh}(C)]\simeq [{\Delta }^{\mathrm{op}},[C^{\mathrm{op}},C]]$$A \in SPSh(C) := [\Delta^{op}, PSh(C)] \simeq [\Delta^{op}, [C^{op}, C]]

  that satisfies degreewise the sheaf condition;

• an SSet-valued presheaf

  $$A\in \mathrm{PSh}(C,\mathrm{SSet}):=[C^{\mathrm{op}},\mathrm{SSet}]\simeq [C^{\mathrm{op}},[{\Delta }^{\mathrm{op}},\mathrm{Set}]]$$A \in PSh(C,SSet) := [C^{op}, SSet] \simeq [C^{op}, [\Delta^{op}, Set]]

  which, when regarded under the equivalence

  $$\mathrm{PSh}(C,\mathrm{SSet})\simeq \mathrm{SPSh}(C)$$PSh(C,SSet) \simeq SPSh(C)

  is degreewise a sheaf.

Model structure

The Jardine-local model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves. This restriction is a Quillen equivalence, so that equipped with this model structure $\mathrm{SSh}(C)$ is a model for the hypecomplete (infinity,1)-topos? over the site $C$.

References

A discussion of the homotopy theory of simplicial objects in toposes using Cisinski model structures is in

• Garth Warner, Homotopical topos theory (pdf)

The last part of

• Peter Johnstone, Sketches of an Elephant

is announced to be about simplicial objects in toposes, but that part does not exist yet.

For more see at simplicial presheaf and model structure on simplicial presheaves.