nLab representable functor theorem

The representable functor theorem

The representable functor theorem


The representable functor theorem states that:

Dually, the corepresentable functor theorem states that:

As representable functors are closely connected to adjoint functors, this theorem is essentially equivalent to the adjoint functor theorem and to theorems guaranteeing the existence of limits.

Specifically, assuming that CC is copowered over Set (in particular, this is true if CC is cocomplete), a functor CSetC\to Set is a right adjoint functor if and only if it is representable, in which case the left adjoint functor SetCSet\to C sends the singleton set to the representing object, and more generally a set XX to the copower of XX with the representing object.


Last revised on October 31, 2023 at 17:01:28. See the history of this page for a list of all contributions to it.