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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).

LSpice
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Arshak Aivazian
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    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. – მამუკა ჯიბლაძე Dec 15 '22 at 05:45
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    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. – Robert Furber Dec 15 '22 at 07:22
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    I guess you have to decide what your question is (currently, the title asks a different one than the one in the body): a characterization (didn't you already find this?) or a few examples (which were already given above - which then may be posted as answers). – Martin Brandenburg Dec 15 '22 at 07:39
  • When I asked the question, it seemed to me that such categories practically did not exist and such a title seemed appropriate (I expected an answer like "the only such categories are: "). From the characterization I wrote, it is not clear how many such categories there are. I'll edit the question and title now, thanks. – Arshak Aivazian Dec 15 '22 at 08:06
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    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. – Simon Henry Dec 15 '22 at 08:46
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    Another example: The category of suplattice. It is monadic (with the covariant power set mona) and it is equivalent to its opposite category. – Simon Henry Dec 15 '22 at 08:48
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    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. – Leonid Positselski Dec 15 '22 at 09:44
  • Can it happen that 2,3,4 hold and 1 fails? – მამუკა ჯიბლაძე Dec 16 '22 at 06:55
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    (More precisely, can it happen that there are objects satisfying 2,3,4 but there are no objects satisfying all four conditions?) – მამუკა ჯიბლაძე Dec 16 '22 at 07:12
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    @მამუკაჯიბლაძე No, 4 implies 1: if distinct $f,g:X\to Y$ were identified by $\mathrm{Hom}(P,-)$ then the map in 4 could not be epi, let alone regular epi. – Kevin Carlson Dec 16 '22 at 19:31
  • @KevinArlin Right, thanks. So a monadic generator is the same as projective generator? – მამუკა ჯიბლაძე Dec 16 '22 at 21:02
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    @მამუკაჯიბლაძე Yes, well, a regular projective generator admitting copowers. – Kevin Carlson Dec 16 '22 at 21:23
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    In fact (co)completeness is necessary for monadicity anyway, so then equivalent conditions are 2, 4 and that the category is cocomplete – მამუკა ჯიბლაძე Dec 17 '22 at 05:49
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    @მამუკაჯიბლაძე True as far as it goes, although you'd almost never already have cocompleteness established before you could prove monadicity. And categories of algebras over other cocomplete bases than Set need not be cocomplete. – Kevin Carlson Dec 17 '22 at 20:24
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    Let me also note that "bimonadic" might be understood as simultaneously monadic and comonadic, which is, by the way, also an interesting class of categories to consider. – მამუკა ჯიბლაძე Dec 18 '22 at 06:42
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    Since the category needs to be Barr-exact anyway, (4) is equivalent to (1) because every epimorphism is regular. So $C$ is bimonadic iff it is Barr-exact and Barr-coexact, complete and cocomplete, and has a projective generator and an injective cogenerator. – Tim Campion Dec 23 '22 at 23:09
  • For example, every Grothendieck topos with a projective generator is bimonadic. – Tim Campion Dec 23 '22 at 23:18
  • @TimCampion Do you know whether these are more general than presheaf toposes? – მამუკა ჯიბლაძე Dec 26 '22 at 10:24
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    @მამუკაჯიბლაძე Hmmm... I want to say maybe sheaves on a zero-dimensional space? But I'm confused -- every presheaf topos has enough projectives, but not necessarily a projective generator. For instance, the coproduct $C$ of the representables may fail to be a generator -- there may be presheaves $X$ which admit no maps from some particular representable, and hence no maps from $C$. – Tim Campion Dec 26 '22 at 16:47
  • @TimCampion Thank you, by some reason this circumstance never occurred to me! All one might hope for is monadicity over a slice $\operatorname{Set}/S$, then. As for zero-dimensionals, here Benjamin Steinberg mentions the abelian case. There, categories of unital modules over non-unital rings with local units appear. I wonder if in the set-theoretic case something like this happens with semigroups in place of rings... – მამუკა ჯიბლაძე Dec 26 '22 at 17:30
  • Actually I just recalled the paper "Collapsed toposes and cartesian closed varieties" by Johnstone (J. Algebra 1990) where, among many other relevant things, he in Theorem 1.4 characterizes toposes with a set of projective generators, as those for which the full subcategory of objects with global support, together with the initial object, is monadic over Set. In Example 8.8 he also considers zero-dimensional spaces. – მამუკა ჯიბლაძე Dec 26 '22 at 18:23
  • And I also recalled that presheaf toposes are characterized as ones having indecomposable projective generators. Whereas the Jónsson-Tarski topos is monadic and, if I am not mistaken, does not possess any nontrivial indecomposable objects – მამუკა ჯიბლაძე Dec 26 '22 at 19:31

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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.

Tim Campion
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    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. – Tim Campion Dec 24 '22 at 18:13
  • I'll accept this answer and a special thanks to everyone for your examples in the comments. It was very helpful for me. – Arshak Aivazian Jan 09 '23 at 05:41