I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor
$$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$
is representable. Is there a name for such categories?
I thought they might be equivalent to some existing notion such as locally presentable categories or total categories, but I can't prove the equivalence in either case.
Best Answer
Without the cocompleteness condition, these were studied in G. M. Kelly's paper "A survey of totality for enriched and ordinary categories" under the name compact categories, and in this mathoverflow question under the much better name saft categories (after the Special Adjoint Functor Theorem).
In that question, Theo Johnson-Freyd gives the following equivalent characterisations:
Theorem. Let $\mathbf{C}$ be a locally small category. Then the following are equivalent:
Every continuous functor $\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is representable.
For all locally small $\mathbf{D}$, every cocontinuous functor $\mathbf{C}\to\mathbf{D}$ has a right adjoint.
Proof. (1) $\Rightarrow$ (2) Let $F:\mathbf{C}\to\mathbf{D}$ be cocontinuous. For $d\in\mathbf{D}$ the functor $\mathrm{Hom}(F(-),d):\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is continuous, and hence representable by an object $G(d)\in\mathbf{C}$. Then given any $c\in\mathbf{C}$ we have $\mathrm{Hom}(F(c),d)\simeq\mathrm{Hom}(c,G(d))$, which [by Categories for the Working Mathematician IV.2.ii] is enough to establish that $F$ has a right adjoint agreeing with $G$ on objects.
(2) $\Rightarrow$ (1) Let $F:\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ be continuous. Then $F^\mathrm{op}:\mathbf{C}\to\mathbf{Set}^\mathrm{op}$ is cocontinuous and $\mathbf{Set}^\mathrm{op}$ is locally small, so $F^\mathrm{op}$ has a right adjoint $G$. Then $F$ is represented by $G(1)$ since $F(c)\simeq\mathrm{Hom}_\mathbf{Set}(1,F(c))\simeq\mathrm{Hom}_{\mathbf{Set}^\mathrm{op}}(F(c),1)\simeq\mathrm{Hom}_\mathbf{C}(c,G(1))$. $\square$
They also point out that every saft category must be complete (since for any small diagram $F:\mathbf{K}\to\mathbf{C}$ the functor $\lim_{k\in K}\mathrm{Hom}(-,F(k)):\mathbf{C}^\mathrm{op}\to\mathbf{Set}$ is continuous, and hence representable by an object of $\mathbf{C}$ which is therefore the limit of $F$). Interestingly they don't have to be cocomplete, a property normally needed for the SAFT.