nLab corepresentable functor

A functor CSetC \to Set is corepresentable if it is (isomorphic to) a functor of the form Hom C(c,)Hom_C(c,-) for some object cCc\in C.

This is equivalently a representable functor defined on the opposite category C opC^{op}. Often no terminological distinction is made between representable and corepresentable ones (both being called simply “representable”), since a functor CSetC\to Set can only be “corepresentable” while a functor C opSetC^{op}\to Set can only be “representable”.

Particularly in the study of moduli problems, there is also the notion of corepresentable contravariant functors. A functor F:C opSetF : C^op \to Set is corepresentable if and only if there exists an object XX in CC and a morphism Fh XF \to h_X such that for any object TT in CC, the canonical map Hom C(X,T)Hom PSh(C)(h X,h T)Hom PSh(C)(F,h T)Hom_C(X,T) \to Hom_{\PSh(C)}(h_X, h_T) \to Hom_{PSh(C)}(F, h_T) is bijective.

Last revised on June 14, 2020 at 05:45:51. See the history of this page for a list of all contributions to it.