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Simplicial presheaf

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In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.

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Example: Let us consider, say, the étale site of a scheme S. Each U in the site represents the presheaf Hom ⁡ ( − , U ) . Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).

Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf B G . For example, one might set B GL = lim → ⁡ B G L n . These types of examples appear in K-theory.

If f : X → Y is a local weak equivalence of simplicial presheaves, then the induced map Z f : Z X → Z Y is also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves π ∗ F of F is defined as follows. For any f : X → Y in the site and a 0-simplex s in F(X), set ( π 0 pr F ) ( X ) = π 0 ( F ( X ) ) and ( π i pr ( F , s ) ) ( f ) = π i ( F ( Y ) , f ∗ ( s ) ) . We then set π i F to be the sheaf associated with the pre-sheaf π i pr F .

Model structures

The category of simplicial presheaves on a site admits many different model structures.

Some of them are obtained by viewing simplicial presheaves as functors

S o p → Δ o p S e t s

The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps

F → G

such that

F ( U ) → G ( U )

is a weak equivalence / fibration of simplicial sets, for all U in the site S. The injective model structure is similar, but with weak equivalences and cofibrations instead.

Stack

A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H →X, the canonical map

F ( X ) → holim ⁡ F ( H n )

is a weak equivalence as simplicial sets, where the right is the homotopy limit of

[ n ] = { 0 , 1 , … , n } ↦ F ( H n ) .

Any sheaf F on the site can be considered as a stack by viewing F ( X ) as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly F ↦ π 0 F .

If A is a sheaf of abelian group (on the same site), then we define K ( A , 1 ) by doing classifying space construction levelwise (the notion comes from the obstruction theory) and set K ( A , i ) = K ( K ( A , i − 1 ) , 1 ) . One can show (by induction): for any X in the site,

H i ⁡ ( X ; A ) = [ X , K ( A , i ) ]

where the left denotes a sheaf cohomology and the right the homotopy class of maps.

References

Simplicial presheaf Wikipedia
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