The representable and corepresentable functor theorems are simple consequences of the general adjoint functor theorem and the following observations:
Suppose is a cocomplete category. A functor is representable if and only if it has a left adjoint . In this case, is the representing object. More generally, is the product of copies of , which exists.
Suppose is a complete category. A functor is corepresentable if and only if it has a left adjoint . In this case, is the corepresenting object. More generally, is the coproduct of copies of , which exists.
The representable functor theorem states that:
The latter condition means the following: there is a set , an -indexed family of objects , and an -indexed family of elements such that for every object and there is and such that .
Dually, the corepresentable functor theorem states that:
The latter condition means the following: there is a set , an -indexed family of objects , and an -indexed family of elements such that for every object and there is and such 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 is copowered over Set (in particular, this is true if is cocomplete), a functor is a right adjoint functor if and only if it is representable, in which case the left adjoint functor sends the singleton set to the representing object, and more generally a set to the copower of with the representing object.
Bodo Pareigis, Thm. 1 on p. 109 in: Categories and Functors, Pure and Applied Mathematics 39, Academic Press (1970) [doi:10.5282/ubm/epub.7244, pdf]
Saunders MacLane, p. 118 (2nd ed: 130) in: Categories for the Working Mathematician, Graduate texts in mathematics 5, Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Last revised on April 20, 2025 at 23:18:07. See the history of this page for a list of all contributions to it.