Strongly compact categories (reference request)

Question

The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called compact when every functor $\mathcal{C} \to \mathcal{D}$ into any category $\mathcal{D}$ which preserves all (possibly large!) colimits is a left adjoint. Equivalently, every presheaf $\mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ preserving all (possibly large!) limits is representable.

Personally, I find large limits and colimits a bit awkward (but maybe you can convince me why they are important?), and in my research I need a property which is perhaps stronger: Let's call a category $\mathcal{C}$ strongly compact when every functor $\mathcal{C} \to \mathcal{D}$ which preserves small colimits is a left adjoint (and hence preserves all colimits). This is stronger because we deal with a a priori larger class of functors. Equivalently, every presheaf on $\mathcal{C}$ preserving small limits is representable.

I have done quite a bit of literature research, but did not find this property elsewhere. But it is very natural, in particular in the context of the special adjoint functor theorem, which says that every wellcopowered cocomplete category with a generator is strongly compact. So I suspect that this notion appears somewhere? If not, what name do you suggest? An alternative would be "realized-sketchable" since I can show that a category is strongly compact if and only if it is the category of models of a (possibly large) realized limit sketch for which every model is small.

*Edit. In my paper I am using the name strong compact.

$\color{red}{^1}$ J. R. Isbell, Small subcategories and completeness, Mathematical systems theory 2.1 (1968): 27-50

$\color{red}{^2}$ R. Börger, W. Tholen, M. B. Wischnewsky, H. Wolff, Compact and hypercomplete categories, Journal of Pure and Applied Algebra 21.2 (1981): 129-144

Answer 1

Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous and satisfies the solution set condition. Therefore, a small-cocomplete category $\mathcal C$ is strongly compact in your sense if and only if every small-cocontinuous functor with domain $\mathcal C$ satisfies the solution set condition. These categories were studied in Ulmer's The adjoint functor theorem and the Yoneda embedding (see Theorem 13, for instance), but were not given a name. Ulmer did not study the variant where $\mathcal C$ is not required to be small-cocomplete.

The relationship to compactness in the sense of Isbell is described in Theorem 8 (where functors preserving all large colimits are called supercocontinuous).

(Thanks to Ivan Di Liberti for pointing out a mistake in the original answer.)

Answer 2

In Adjoints to functors from categories of algebras Rattray defines that a category $\mathcal{A}$ has LAP (left adjoint property) when every continuous functor on $\mathcal{A}$ has a left adjoint, or equivalently every continuous functor $\mathcal{A} \to \mathbf{Set}$ is representable. This is exactly the dual notion to what I called "strongly compact". So Rattray would probably call this property RAP (right adjoint property). He shows that every monadic category over a RAP category is again a RAP category. (He proves the dual version: every comonadic category over a LAP category is a LAP category.)

It is a bit confusing that it is claimed in the mentioned paper "Compact and hypercomplete categories" by R. Börger, W. Tholen, M. B. Wischnewsky, H. Wolff that Rattray actually proves that every monadic category over a compact category is again compact: compactness involves hypercontinuous functors. But probably Rattray's proof can be used in both settings.