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presheaf_._topology [2014/10/30 10:26]
nikolaj
presheaf_._topology [2014/10/30 10:26]
nikolaj
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   * 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). ​
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