# nLab representable functor theorem

The representable functor theorem

category theory

# The representable functor theorem

## Statement

The representable functor theorem states that:

• A presheaf $C^{op} \to Set$ on a cocomplete category $C$ is representable (i.e. of the form $C({-}, c)$ for some object $c \in C$) if and only if it sends colimits in $C$ to limits in $Set$ and has a solution set.

Dually, the corepresentable functor theorem states that:

• A copresheaf $C \to Set$ on a complete category $C$ is corepresentable (i.e. of the form $C(c, {-})$ for some object $c \in C$) if and only if it is continuous and has a solution set.

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

## References

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