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presheaf_category [2014/10/30 10:24]
nikolaj
presheaf_category [2015/02/21 12:10] (current)
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]]
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