Special adjoint functor theorem

Remarks

Here a set $Q$ of objects is cogenerating when, for any two different morphisms $f, g \colon a \to b$, some $q \colon b \to x$ with $x \in Q$ has $q f \neq q g$. This is the dual of a generator. The proof constructs an initial object of each comma category $c \downarrow G$. Let $P$ be the product of copies of $x$, one for each $x \in Q$ and each morphism $c \to G x$. Then one intersects all subobjects $S$ of $P$ such that the induced map $c \to G P$ factors through $G S$. Well-poweredness makes this intersection a small limit.

Example: the category of compact Hausdorff spaces is complete and well-powered, and the unit interval $[0, 1]$ is a cogenerator. The inclusion into topological spaces preserves limits. So it has a left adjoint, the Stone–Čech compactification.

Going deeper

For a student

Compared with Freyd's general theorem, the solution set condition is replaced by well-poweredness and a cogenerating set in $\mathcal{D}$. These conditions concern only $\mathcal{D}$, so they are checked once for all functors out of $\mathcal{D}$. The dual theorem gives right adjoints of colimit-preserving functors out of a cocomplete, co-well-powered category with a generating set. For instance, every colimit-preserving functor from a category of presheaves on a small category has a right adjoint.