nLab
simplicial sheaf

Definition

A simplicial sheaf A is equivalently

  • a simplicial object

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

    in a category of sheaves Sh(C) for some site C;

  • a simplicial presheaf

    ASPSh(C):=[Δ op,PSh(C)][Δ op,[C 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

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

    which, when regarded under the equivalence

    PSh(C,SSet)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 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.