Properties
Note that a functor of the form
- Hom(–, C) : Cop → Set
is a presheaf; likewise, Hom(C, –) is a copresheaf.
A functor F : C → Set that is naturally isomorphic to Hom(C, –) is called a representable functor or sometimes a representable copresheaf; likewise, a contravariant functor equivalent to Hom(–, C) might be called corepresentable.
Note that Hom(–, –) : Cop × C → Set is a profunctor, and, specifically, it is the identity profunctor
- ,
The internal hom functor preserves limits; that is, sends limits to limits, while sends limits to colimits. In a certain sense, this can be taken as the definition of a limit or colimit.
Read more about this topic: Hom Functor
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)