Homotopy sheaf

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.

Example: Let us consider, say, the étale site of a scheme S. Each U in the site represents the presheaf ${\displaystyle \operatorname {Hom} (-,U)}$. 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 ${\displaystyle BG}$. For example, one might set ${\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }$. These types of examples appear in K-theory.

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