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| ==== Discussion ==== | ==== Discussion ==== | ||
| - | It's worth stating that set-valued functors $F$ in general are often about functions in the sense that if $A:\mathrm{Ob}_{\bf C}$, then $FA$ is often some function space whose definition involves $A$. The co- and contravariant hom-functors $\mathrm{Hom}(B,-)$ and $\mathrm{Hom}(-,B)$ are two examples. (But that's of course no rule - the forgetful functors are also relevant examples for set-valued functors!) | + | The co- and contravariant hom-functors $\mathrm{Hom}(B,-)$ and $\mathrm{Hom}(-,B)$ are maybe the most natural functors. While forgetful functors are other examples of covariant set-valued functors, covariant functors very often have to do with function spaces. (Once we pass from presheaves to sheaves by adding some more "topological requirements", this becomes a theorem: sheaves can always be viewed as evaluating to collections of function spaces.) |
| === Example === | === Example === | ||
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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. | ||
