Brown's Representability Theorem

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

More specifically, we are given

F: Hotcop → Set,

and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.

Read more about Brown's Representability Theorem:  Brown Representability Theorem For CW Complexes, Variants

Famous quotes containing the words brown and/or theorem:

    His reversed body gracefully curved, his brown legs hoisted like a Tarentine sail, his joined ankles tacking, Van gripped with splayed hands the brow of gravity, and moved to and fro, veering and sidestepping, opening his mouth the wrong way, and blinking in the odd bilboquet fashion peculiar to eyelids in his abnormal position. Even more extraordinary than the variety and velocity of the movements he made in imitation of animal hind legs was the effortlessness of his stance.
    Vladimir Nabokov (1899–1977)

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)