User:TakuyaMurata/Sandbox/Sheaf (infinity category)

Given an ∞-category C that admits limits and given the category Sch of quasi-projective schemes over a field k equipped with the étale topology, a functor F: Sch^op →C is called a sheaf if

• (1) The F of the empty set is the terminal object of C.
• (2) For any increasing sequence $U_{i}$ of open subsets with union U, the canonical map $F(U)\to \varprojlim F(U_{i})$ is an equivalence.
• (3) $F(U\cap V)$ is the pullback of $F(U\cup V)\to F(U)$ and $F(U\cup V)\to F(V)$.

If C is the nerve of a category, then the notion reduces to the usual one. The sheaves form a full subcategory of Fun(Sch^op, C). The left adjoint of this inclusion of sheaves is called the sheafification functor.

Examples

• Given a finite abelian group M, let $c_{M}$ denote the constant presheaf given by M; i.e., $c_{M}(X)$ consists of locally constant functions X →M. Unlike the classical case, it is not a sheaf. The sheafification of $c_{M}$ is then denoted by $C^{*}(-;M)$. By Dold–Kan, $C^{*}(X;M)$ can be identified, up to equivalence, with the injective resolution of M applied to X; in other words, the cohomology of $C^{*}(X;M)$ is the usual étale cohomology of X with coefficients in the constant étale sheaf M.