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| A [[sheaf]] $F$ is particular kind of contravariant functor which helps to capture sections. The object image of a sheaf of a an open set $FU$ is a local function space. | A [[sheaf]] $F$ is particular kind of contravariant functor which helps to capture sections. The object image of a sheaf of a an open set $FU$ is a local function space. | ||
| - | === Example === | + | === Examples === |
| * Consider the sheaf $C^\infty$ of all smooth functions over $X$. If $U$ is an open set of $X$, then the objects $C^\infty U:=\{f:U\to\mathbb R\ |\ f\dots\text{smooth} \}$ is the restriction of such functions to $U$. | * Consider the sheaf $C^\infty$ of all smooth functions over $X$. If $U$ is an open set of $X$, then the objects $C^\infty U:=\{f:U\to\mathbb R\ |\ f\dots\text{smooth} \}$ is the restriction of such functions to $U$. | ||
| * Any continuous function Take $X=\mathbb{C}\setminus\{0\}$, $E=\mathbb C$, and $p=\exp$. Define a the sheaf $\Gamma(E/X)$ via $\Gamma(E/X)U:=\{s:U\to E\ |\ p\circ s=\mathrm{id}_U\}$. Then $\Gamma(E/X)U$ is the set of branches of the logarithm on $U$. | * Any continuous function Take $X=\mathbb{C}\setminus\{0\}$, $E=\mathbb C$, and $p=\exp$. Define a the sheaf $\Gamma(E/X)$ via $\Gamma(E/X)U:=\{s:U\to E\ |\ p\circ s=\mathrm{id}_U\}$. Then $\Gamma(E/X)U$ is the set of branches of the logarithm on $U$. | ||
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| + | == /Examples == | ||
| Sheaf theory is used for discussing the idea of analytical continuation and applying it too subjects which don't necessarily have to do with complex numbers. The image of the functor can contain other objects than sections (e.g. vector bundles as in K-theory, or spaces of differential forms, etc.) and the category itself doesn't have to be a classical topological space (see Grothendieck topology). | Sheaf theory is used for discussing the idea of analytical continuation and applying it too subjects which don't necessarily have to do with complex numbers. The image of the functor can contain other objects than sections (e.g. vector bundles as in K-theory, or spaces of differential forms, etc.) and the category itself doesn't have to be a classical topological space (see Grothendieck topology). | ||
