# Differences

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sheaf [2014/10/29 10:15] nikolaj |
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A seperated presheaf is a sheaf if such a collection of selected to-be-partions-of-a-section $S(V)$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$. | A seperated presheaf is a sheaf if such a collection of selected to-be-partions-of-a-section $S(V)$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$. | ||

So the postulate says that the image of $F$ contains all sections which can arise from gluing together other available sections. | So the postulate says that the image of $F$ contains all sections which can arise from gluing together other available sections. | ||

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+ | === Example === | ||

+ | From the Wikipedia page below: | ||

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+ | "Any continuous map of topological spaces determines a sheaf of sets. Let f : Y → X be a continuous map. We define a sheaf Γ(Y/X) on X by setting Γ(Y/X)(U) equal to the sections U → Y, that is, Γ(Y/X)(U) is the set of all continuous functions s : U → Y such that f ∘ s = id_U. Restriction is given by restriction of functions. This sheaf is called the sheaf of sections of f, and it is especially important when f is the projection of a fiber bundle onto its base space. Notice that if the image of f does not contain U, then Γ(Y/X)(U) is empty. For a concrete example, take X = C \ {0}, Y = C, and f(z) = exp(z). Γ(Y/X)(U) is the set of branches of the logarithm on U." | ||

=== Reference === | === Reference === |