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presheaf_category [2014/10/30 10:24] nikolaj |
presheaf_category [2015/02/21 12:10] nikolaj |
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| | @#FFBB00: definiendum | @#FFBB00: ${\bf Set}^{{\bf C}^\mathrm{op}}$ | | | @#FFBB00: definiendum | @#FFBB00: ${\bf Set}^{{\bf C}^\mathrm{op}}$ | | ||
| - | ==== Discussion ==== | + | ----- |
| 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.) | 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.) | ||
| === Reference === | === Reference === | ||
| Wikipedia: | Wikipedia: | ||
| - | [[http://en.wikipedia.org/wiki/Yoneda_lemma|Yoneda lemma]] | + | [[http://en.wikipedia.org/wiki/Yoneda_lemma|Yoneda lemma]], |
| + | [[http://en.wikipedia.org/wiki/Functor_category|Functor category]] | ||
| - | === Reference === | + | ----- |
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Functor_category|Functor category]] | + | |
| - | ==== Parents ==== | + | |
| === Subset of === | === Subset of === | ||
| [[Functor category]] | [[Functor category]] | ||
