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 $\operatorname(-, U)$. 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 $B\operatorname = \varinjlim B\operatorname$. 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 $\mathbb f: \mathbb X \to \mathbb Y$ is also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf

Let ''F'' be a simplicial presheaf on a site. The homotopy sheaves $\pi_* F$ of ''F'' are defined as follows. For any $f:X \to Y$ in the site and a 0-simplex ''s'' in ''F''(''X''), set $(\pi_0^\text F)(X) = \pi_0 (F(X))$ and $(\pi_i^\text (F, s))(f) = \pi_i (F(Y), f^*(s))$. We then set $\pi_i F$ to be the sheaf associated with the pre-sheaf $\pi_i^\text F$.

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
:$S^ \to \Delta^ Sets$
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
:$\mathcal F \to \mathcal G$
such that
:$\mathcal F(U) \to \mathcal G(U)$
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
:$F(X) \to \operatorname F(H_n)$
is a weak equivalence as simplicial sets, where the right is the homotopy limit of
: = \ \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,
:\operatorname^i(X; A) = , K(A, i)/math>
where the left denotes a sheaf cohomology and the right the homotopy class of maps.

See also

*cubical set
*N-group (category theory)

Notes

Further reading

*Konrad Voelkel
Model structures on simplicial presheaves

References

*
*
*B. Toën
Simplicial presheaves and derived algebraic geometry
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External links

J.F. Jardine's homepage

Homotopy theory
Simplicial sets
Functors