Category Theory, Representable

Representable

A concrete category is representable if there is a reference object O and a specific element e in O, such that for any other object X in the category, a bijection exists between morphisms from O to X and the elements of X, given by the image of e in X. Thus the image of e in X determines uniquely (represents) the morphism.

Any subcategory containing O remains representable.

The simplest representable categories are certain monoids, i.e. one object. Consider the ring endomorphisms of Z_n. If you know where 1 goes, you have the entire endomorphism in hand, and there is a ring endomorphism for every element of Z_n.

The category of sets is representable, in a rather trivial way. Let O be the set containing e, and any function from O into X is the image of e. Most of the categories you know are not representable, unless you select a rather trivial reference object O, as described above.