Simplicial presheaf

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.

Examples

Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf $\mathrm{Hom}(-,U)$. 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 $BG$. For example, one might set $B\mathrm{GL}=\underrightarrow{\lim }B\mathrm{G}{\mathrm{L}}_{\mathrm{n}}$. These types of examples appear in K-theory.

If $f:X\to Y$ is a local weak equivalence of simplicial presheaves, then the induced map $\mathbb{Z}f:\mathbb{Z}X\to \mathbb{Z}Y$ is also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves ${\pi }_{\ast }F$ of F are defined as follows. For any $f:X\to Y$ in the site and a 0-simplex s in F(X), set $({\pi }_0^{\text{pr}}F)(X)={\pi }_0(F(X))$ and $({\pi }_i^{\text{pr}}(F,s))(f)={\pi }_i(F(Y),f^{\ast }(s))$. We then set ${\pi }_{i}F$ to be the sheaf associated with the pre-sheaf ${\pi }_i^{\text{pr}}F$.

Model structures

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

Some of them are obtained by viewing simplicial presheaves as functors

$S^{op}\to {\mathrm{\Delta }}^{op}Sets$

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

$ℱ\to 𝒢$

such that

$ℱ(U)\to 𝒢(U)$

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

$F(X)\to \mathrm{holim}F(H_n)$

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

$[n]=\{0,1,\dots ,n\}\mapsto F(H_n)$.

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 $F\mapsto {\pi }_{0}F$.

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,

${\mathrm{H}}^i(X;A)=[X,K(A,i)]$

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

See also

• cubical set
• N-group (category theory)

Further reading

• Konrad Voelkel, Model structures on simplicial presheaves

References

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• B. Toën, Simplicial presheaves and derived algebraic geometry

External links

• J.F. Jardine's homepage