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:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)