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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. 1, $\mathrm{Set}$, $\mathrm{CompHaus}$, $\mathrm{SupLat}$ are bimonadic.
  2. $C \times D$ 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

  1. $C$ Barr-exact and co(Barr-exact).
  2. $C$ has a monadic generator and a comonadic cogenerator.

A monadic generator is an object $P$ such that

  1. $P$ is a separator (i.e. $\operatorname{Hom}(P, -)$ faithful)
  2. $P$ is projective (that is, $\operatorname{Hom}(P, -)$ preserves epimorphisms)
  3. $P$ has all copowers $\coprod_A P$
  4. 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).

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    $\begingroup$ By a theorem of Paré, for every elementary topos $\mathscr E$ its opposite category is monadic over $\mathscr E$ (it is equivalent to the category of internal complete atomic Heyting algebras in $\mathscr E$). The monad in question is the double powerset monad. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Dec 15, 2022 at 5:45
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    $\begingroup$ I know an answer to the question in the title at least. The category of compact Hausdorff spaces $\mathrm{CHaus}$ is monadic over $\mathrm{Set}$ by the usual forgetful functor and its left adjoint the Stone-Čech compactification. Gelfand duality makes $\mathrm{CHaus}$ dual to the category of commutative unital C$^*$-algebras $\mathrm{CC}^*$. Then the unit ball functor $\mathrm{CC}^* \rightarrow \mathrm{Set}$ is also monadic, in fact $\aleph_1$-accessibly so - the free commutative C$^*$-algebra on $X$ is $C(\mathbb{D}^X)$ where $\mathbb{D}$ is the closed complex unit disc. $\endgroup$
    – Robert Furber
    Commented Dec 15, 2022 at 7:22
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    $\begingroup$ Interestingly - because as you mentioned, except for complete posets $C$ and $C^{op}$ are never both locally presentable. If $C$ and $C^{op}$ are monadic than one of the two monads has to be a non-accessible monad. $\endgroup$
    – Simon Henry
    Commented Dec 15, 2022 at 8:46
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    $\begingroup$ Another example: The category of suplattice. It is monadic (with the covariant power set mona) and it is equivalent to its opposite category. $\endgroup$
    – Simon Henry
    Commented Dec 15, 2022 at 8:48
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    $\begingroup$ Any abelian category with coproducts and a projective generator is monadic. Hence any abelian category with products, coproducts, projective generator, and injective cogenerator is bimodanic. For example, the category of modules over an arbitrary ring is bimonadic. $\endgroup$
    – Leonid Positselski
    Commented Dec 15, 2022 at 9:44

1 Answer 1

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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:

  1. $\mathcal C$ is monadic over $Set$;
  2. $\mathcal C$ is Barr-exact and has a projective generator;
  3. $\mathcal C$ 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.

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    $\begingroup$ Maybe it's worth noting the following consequence of the Corollary: if $\mathcal C$ is already known to be monadic over $Set$, then in order to show that $\mathcal C$ is bimonadic, you need only check that $\mathcal C$ has an injective cogenerator. $\endgroup$
    – Tim Campion
    Commented Dec 24, 2022 at 18:13
  • $\begingroup$ I'll accept this answer and a special thanks to everyone for your examples in the comments. It was very helpful for me. $\endgroup$
    – Arshak Aivazian
    Commented Jan 9, 2023 at 5:41

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