How many categories $C$ are there such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$? In this question, bimonadic category is a category $C$  such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$.

How many bimonadic categories are there? Can we classify them all?

Currently (updated):

*

*1, $\mathrm{Set}$, $\mathrm{CompHaus}$, $\mathrm{SupLat}$ are bimonadic.

*$C \times D$ is bimonadic if $C$ and $D$ are bimonadic.

By the characterization theorem for monadic categories over $\mathrm{Set}$ (see Borceux, HCA II), this question is equivalent to searching categories $C$ with the following properties

*

*$C$ Barr-exact and co(Barr-exact).

*$C$ has a monadic generator and a comonadic cogenerator.

A monadic generator is an object $P$ such that

*

*$P$ is a separator (i.e. $\operatorname{Hom}(P, -)$ faithful)

*$P$ is projective (that is, $\operatorname{Hom}(P, -)$ preserves epimorphisms)

*$P$ has all copowers $\coprod_A P$

*For any $X \in \operatorname{Ob} C$, the natural morphism $\coprod_{f: P \to X} P \to X$ is a regular epimorphism.

Comonadic cogenerator is a formal dualization of this concept.
This characterization is not an answer to the question, because directly from it it is not clear how to check whether a given category is bimonadic.
P.S. A similar question Can the opposite of an elementary topos be an elementary topos? about toposes states that the opposite category of a locally presentable category is never locally presentable (with the exception of complete posets?), but a monadic category is not necessarily locally presentable (for example, the category of frames).
 A: My comments are overflowing, so let me just record here that if you want minimal, easy-to-check conditions for a category $\mathcal C$ to be monadic over $Set$, then Borceux's Theorem 4.4.5 in the Handbook of Categorical Algebra 2 is not stated optimally, at least if, like me, you're happy to check co/cocompleteness separately. If you go through his proof, you will find that the full strength of the assumption that $\mathcal C$ is Barr-exact is not really used. I believe that Borceux's proof actually shows the following:
Thm: (Borceux) Let $\mathcal C$ be a complete and cocomplete category. Then the following are equivalent:

*

*$\mathcal C$ is monadic over $Set$;

*$\mathcal C$ is Barr-exact and has a projective generator;

*$\mathcal C$ has unique epi-mono factorizations and a projective generator.

Corollary: Let $\mathcal C$ be a complete and cocomplete category. Then $\mathcal C$ is bimonadic if and only if it has unique epi-mono factorizations as well as a projective generator and an injective cogenerator.

In the interest of looking for more examples of bimonadic categories, I think the above conditions are perhaps easier to check than the conditions that Borceux lists (mostly because thinking about Barr-coexactness gives me a headache, so it's convenient to avoid having to consider it). In the other direction, I suspect it may not be feasible to really write down a list of all bimonadic categories.
