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sheaf [2014/10/29 10:10] nikolaj |
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| ==== Discussion ==== | ==== Discussion ==== | ||
| + | As in [[seperated presheaf]], $t|_V$ denotes the image of $t$ under $F(i)$ with $i:V\to W$. | ||
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| ===Elaboration=== | ===Elaboration=== | ||
| Previously, we defined the 'Locality axiom' which makes a [[presheaf . topology|presheaf]] into a [[seperated presheaf]]. The postulate in this entry is the second axiom which makes a [[seperated presheaf]] into a sheaf. | Previously, we defined the 'Locality axiom' which makes a [[presheaf . topology|presheaf]] into a [[seperated presheaf]]. The postulate in this entry is the second axiom which makes a [[seperated presheaf]] into a sheaf. | ||
| - | In the above, if $V\in C$ are elements of a cover of an open set, the function $S$ selects one section $S(V)$ from every $FV$. | + | **Gluing axiom**: If $V\in C_U$ are elements of a cover $C_U$ of an open set $U$, the function $S$ selects one section $S(V)$ from every $FV$. |
| + | 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. | ||
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| + | === Example === | ||
| + | From the Wikipedia page below: | ||
| - | **Gluing axiom**: Let $V,U$ with $V\subset U$ be open sets. A seperated presheaf is a sheaf if a bunch of chosen to-be-partions-of-a-section $S(V)\in FV$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$. | + | "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." |
| - | The postulate says that the image of $F$ contains contain all sections which can arise from gluing together other available sections. | + | |
| === Reference === | === Reference === | ||
