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homomorphisms of (pre)sheaves

DEFINITION 1.23   Let $ \mathcal F_1$ , $ \mathcal F_2$ be presheaves of modules on a topological space $ X$ . Then we say that a sheaf homomorphism

$\displaystyle \varphi:\mathcal F_1 \to \mathcal F_2
$

is given if we are given a module homomorphism

$\displaystyle \varphi_U: \mathcal F_1(U) \to \mathcal F_2(U)
$

for each open set $ U\subset X$ with the following property hold.
  1. For any open subsets $ V,U \subset X$ such that $ V\subset U$ , we have

    $\displaystyle \rho_{V,U} \circ \varphi_U=\varphi_V\circ \rho_{V,U}.
$

(The property is also commonly referred to as ``$ \varphi$ commutes with restrictions''.)

DEFINITION 1.24   A homomorphism of sheaves is defined as a homomorphism of presheaves.



2007-12-11