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presheaf_category [2014/07/19 18:38] nikolaj |
presheaf_category [2014/07/21 21:01] nikolaj |
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| Consider $\mathbb C\setminus\{0\}$ and all its open subsets as objects in a category with single arrows representing the inclusions. Using the exponential map $\mathrm{exp}$, we define the sheaf $F$ mapping an open set $U\subseteq \mathbb C$ to the set $FU$ of all sections $s$ with $\mathrm{exp}\circ s=\mathrm{id}$ and the arrows maps being restrictions of sections to the smaller open set. This way we have constructed the sheaf of all branches of the logarithm! The maximal analytical continuation can now be characterized as the section over the largest open set, where largeness is determined by climbing up the arrows. | Consider $\mathbb C\setminus\{0\}$ and all its open subsets as objects in a category with single arrows representing the inclusions. Using the exponential map $\mathrm{exp}$, we define the sheaf $F$ mapping an open set $U\subseteq \mathbb C$ to the set $FU$ of all sections $s$ with $\mathrm{exp}\circ s=\mathrm{id}$ and the arrows maps being restrictions of sections to the smaller open set. This way we have constructed the sheaf of all branches of the logarithm! The maximal analytical continuation can now be characterized as the section over the largest open set, where largeness is determined by climbing up the arrows. | ||
| - | 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). |
| === Reference === | === Reference === | ||
