nLab simplicial sheaf

Definition

A simplicial sheaf $A$ is equivalently

• a simplicial object

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

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

• a simplicial presheaf

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

  that satisfies degreewise the sheaf condition;

• an SSet-valued presheaf

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

  which, when regarded under the equivalence

  $$PSh(C,SSet) \simeq SPSh(C)$$

  is degreewise a sheaf.

Joyal 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 $SSh(C)$ is a model for the hypercomplete (infinity,1)-topos over the site $C$.

Historically, Joyal constructed his model structure on simplicial sheaves first and Jardine later constructed his model structure on simplicial presheaves.

Local projective model structure

Blandner proved that the category of simplicial sheaves admits a model structure whose weak equivalences are the same as in the Joyal model structure and acyclic fibrations are the projective (i.e., degreewise) acyclic fibrations.

Related concepts

• simplicial presheaf

References

The original construction due to Joyal in 1984 is in

• André Joyal, Lettre d’André Joyal à Alexandre Grothendieck, April 11, 1984, PDF.

Discussion of the homotopy theory of simplicial objects in toposes using Cisinski model structures:

• Garth Warner, Homotopical Topos Theory (2012) [pdf, 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.