Relation To Universal Morphisms and Adjoints
The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors.
Let G : D → C 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 : C → D 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 proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“You see, I am alive, I am alive
I stand in good relation to the earth
I stand in good relation to the gods
I stand in good relation to all that is beautiful
I stand in good relation to the daughter of Tsen-tainte
You see, I am alive, I am alive”
—N. Scott Momaday (b. 1934)
“The philosopher is like a man fasting in the midst of universal intoxication. He alone perceives the illusion of which all creatures are the willing playthings; he is less duped than his neighbor by his own nature. He judges more sanely, he sees things as they are. It is in this that his liberty consistsin the ability to see clearly and soberly, in the power of mental record.”
—Henri-Frédéric Amiel (18211881)