Representable Functor - Relation To Universal Morphisms and Adjoints

Relation To Universal Morphisms and Adjoints

The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors.

Let G : DC be a functor and let X be an object of C. Then (A,φ) is a universal morphism from X to G if and only if (A,φ) is a representation of the functor HomC(X,G–) from D to Set. It follows that G has a left-adjoint F if and only if HomC(X,G–) is representable for all X in C. The natural isomorphism ΦX : HomD(FX,–) → HomC(X,G–) yields the adjointness; that is

is a bijection for all X and Y.

The dual statements are also true. Let F : CD be a functor and let Y be an object of D. Then (A,φ) is a universal morphism from F to Y if and only if (A,φ) is a representation of the functor HomD(F–,Y) from C to Set. It follows that F has a right-adjoint G if and only if HomD(F–,Y) is representable for all Y in D.

Read more about this topic:  Representable Functor

Famous quotes containing the words relation to, relation and/or universal:

    The psychoanalysis of individual human beings, however, teaches us with quite special insistence that the god of each of them is formed in the likeness of his father, that his personal relation to God depends on his relation to his father in the flesh and oscillates and changes along with that relation, and that at bottom God is nothing other than an exalted father.
    Sigmund Freud (1856–1939)

    Whoever has a keen eye for profits, is blind in relation to his craft.
    Sophocles (497–406/5 B.C.)

    All men, in the abstract, are just and good; what hinders them, in the particular, is, the momentary predominance of the finite and individual over the general truth. The condition of our incarnation in a private self, seems to be, a perpetual tendency to prefer the private law, to obey the private impulse, to the exclusion of the law of the universal being.
    Ralph Waldo Emerson (1803–1882)