Sheaf (mathematics)
Morphisms

Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which is simply an assignment of outputs to inputs, morphisms of sheaves are also required to be compatible with the local–global structures of the underlying sheaves. This idea is made precise in the following definition.

Let F {\displaystyle F} and G {\displaystyle G} be two sheaves of sets (respectively abelian groups, rings, etc.) on X {\displaystyle X}. A morphism φ: F → G {\displaystyle \varphi:F\to G} consists of a morphism φ U: F (U) → G (U) {\displaystyle \varphi _{U}:F(U)\to G(U)} of sets (respectively abelian groups, rings, etc.) for each open set U {\displaystyle U} of X {\displaystyle X}, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset V {\displaystyle V} of an open set U {\displaystyle U}, the following diagram is commutative.

For example, taking the derivative gives a morphism of sheaves on R {\displaystyle \mathbb {R} }: O R n → O R n − 1. {\displaystyle {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.} Indeed, given an (n {\displaystyle n} -times continuously differentiable) function f: U → R {\displaystyle f:U\to \mathbb {R} } (with U {\displaystyle U} in R {\displaystyle \mathbb {R} } open), the restriction (to a smaller open subset V {\displaystyle V}) of its derivative equals the derivative of f | V {\displaystyle f|_{V}}.

With this notion of morphism, sheaves of sets (respectively abelian groups, rings, etc.) on a fixed topological space X {\displaystyle X} form a category. The general categorical notions of mono-, epi- and isomorphisms can therefore be applied to sheaves.

A morphism φ: F → G {\displaystyle \varphi \colon F\rightarrow G} of sheaves on X {\displaystyle X} is an isomorphism (respectively monomorphism) if and only if there exists an open cover { U α } {\displaystyle \{U_{\alpha }\}} of X {\displaystyle X} such that φ | U α: F (U α) → G (U α) {\displaystyle \varphi |_{U_{\alpha }}\colon F(U_{\alpha })\rightarrow G(U_{\alpha })} are isomorphisms (respectively injective morphisms) of sets (respectively abelian groups, rings, etc.) for all α {\displaystyle \alpha }. These statements give examples of how to work with sheaves using local information, but it's important to note that we cannot check if a morphism of sheaves is an epimorphism in the same manner. Indeed the statement that maps on the level of open sets φ U: F (U) → G (U) {\displaystyle \varphi _{U}\colon F(U)\rightarrow G(U)} are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor—or equivalently, to non-triviality of sheaf cohomology.
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