Definition
Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,–) be the hom functor which maps objects X to the set Hom(A,X).
A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where
- Φ : Hom(A,–) → F
is a natural isomorphism.
A contravariant functor G from C to Set is the same thing as a functor G : Cop → Set and is therefore representable just when it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C.
Read more about this topic: Representable Functor
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)