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: Schop →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(Schop, 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.