nLab
simplicial sheaf

Contents

Definition

A simplicial sheaf AA 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)Sh(C) for some site CC;

  • 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 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 SSh(C)SSh(C) is a model for the hypecomplete (infinity,1)-topos? over the site CC.

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

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.

Revised on February 26, 2013 20:21:24 by Mike Shulman (192.16.204.218)