nLab representable functor theorem

Idea

The representable and corepresentable functor theorems are simple consequences of the general adjoint functor theorem and the following observations:

• Suppose $C$ is a cocomplete category. A functor $F: C^{op}\to Set$ is representable if and only if it has a left adjoint $L$. In this case, $L(\{\ast\})=X$ is the representing object. More generally, $L(S)=X^S$ is the product of $S$ copies of $X$, which exists.

• Suppose $C$ is a complete category. A functor $F:C\to Set$ is corepresentable if and only if it has a left adjoint $L$. In this case, $L(\{\ast\})=X$ is the corepresenting object. More generally, $L(S)=\coprod_S X$ is the coproduct of $S$ copies of $X$, which exists.

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.

The latter condition means the following: there is a set $I$, an $I$-indexed family of objects $X_i\in C$, and an $I$-indexed family of elements $f_i\in F(X_i)$ such that for every object $Y\in C$ and $h\in F(Y)$ there is $k\in I$ and $t\colon Y\to X_i$ such that $h=F(t)(f_i)$.

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.

The latter condition means the following: there is a set $I$, an $I$-indexed family of objects $X_i\in C$, and an $I$-indexed family of elements $f_i\in F(X_i)$ such that for every object $Y\in C$ and $h\in F(Y)$ there is $k\in I$ and $t\colon X_i\to Y$ such that $h=F(t)(f_i)$.

Related facts

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.

Related concepts

• adjoint functor theorem

References

• 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]