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 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 October 15, 2024 at 12:41:25. See the history of this page for a list of all contributions to it.