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 standard model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves, this restriction is a Quillen equivalence and equipped with this model structure $\mathrm{SSh}(C)$ is one of the standard models for ∞-stack (∞,1)-toposes for the site $C$.

Simplicial concrete sheaves

Sometimes it is useful to restrict further to simplicial objects in the category of concrete sheaves on $C$.

For instance for $C$ a suitable site of smooth test spaces a simplicial concrete sheaf is a simplicial diffeological space.