Simplicial sheaves

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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.[1] Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.[2]

Examples

Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf Hom ⁡ ( − , U ) {\displaystyle \operatorname {Hom} (-,U)} {\displaystyle \operatorname {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 {\displaystyle BG} {\displaystyle BG}. For example, one might set B GL = lim → ⁡ B G L n {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} } {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }. These types of examples appear in K-theory.

If f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} is a local weak equivalence of simplicial presheaves, then the induced map Z f : Z X → Z Y {\displaystyle \mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y} {\displaystyle \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 π ∗ F {\displaystyle \pi _{*}F} {\displaystyle \pi _{*}F} of F are defined as follows. For any f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} in the site and a 0-simplex s in F(X), set ( π 0 pr F ) ( X ) = π 0 ( F ( X ) ) {\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))} {\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))} and ( π i pr ( F , s ) ) ( f ) = π i ( F ( Y ) , f ∗ ( s ) ) {\displaystyle (\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))} {\displaystyle (\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))}. We then set π i F {\displaystyle \pi _{i}F} {\displaystyle \pi _{i}F} to be the sheaf associated with the pre-sheaf π i pr F {\displaystyle \pi _{i}^{\text{pr}}F} {\displaystyle \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 o p → Δ o p S e t s {\displaystyle S^{op}\to \Delta ^{op}Sets} {\displaystyle S^{op}\to \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 → G {\displaystyle {\mathcal {F}}\to {\mathcal {G}}} {\displaystyle {\mathcal {F}}\to {\mathcal {G}}}

such that

F ( U ) → G ( U ) {\displaystyle {\mathcal {F}}(U)\to {\mathcal {G}}(U)} {\displaystyle {\mathcal {F}}(U)\to {\mathcal {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 ) {\displaystyle F(X)\to \operatorname {holim} F(H_{n})} {\displaystyle F(X)\to \operatorname {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 ) {\displaystyle [n]=\{0,1,\dots ,n\}\mapsto F(H_{n})} {\displaystyle [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 ) {\displaystyle F(X)} {\displaystyle 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 {\displaystyle F\mapsto \pi _{0}F} {\displaystyle F\mapsto \pi _{0}F}.

If A is a sheaf of abelian group (on the same site), then we define K ( A , 1 ) {\displaystyle K(A,1)} {\displaystyle 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 ) {\displaystyle K(A,i)=K(K(A,i-1),1)} {\displaystyle 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 ) ] {\displaystyle \operatorname {H} ^{i}(X;A)=[X,K(A,i)]} {\displaystyle \operatorname {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

Notes

  1. Toën, Bertrand (2002), "Stacks and Non-abelian cohomology" (PDF), Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory, MSRI
  2. Jardine 2007, §1

Further reading

References