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If $T$ and $S$ are two monads on the category of sets $Set$ with isomorphic algebra categories, i.e. Eilenberg-Moore categories (see https://ncatlab.org/nlab/show/Eilenberg-Moore+category), in what way can we then relate the monads themselves? For example, are they isomorphic as endofunctors? Do they have similar properties?

I ask this question because i am interested in cartesian monads and I would like to know if this property transfers. There are a lot of sources on monads, but I havent seen any that talk about this, so if anyone has a good reference, that is also much appreciated!

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    $\begingroup$ You have to be a bit more specific on what you mean by "isomorphic" here, because there's a chance your question becomes trivial. Do you mean "equivalent as abstract categories", "isomorphic as abstract categories", or "equivalent as categories over $Set$" (you would have to specify what this means, as I don't think there is a standard notion) or finally (the one that makes the question quite easy) "isomorphic as categories over $Set$" ? $\endgroup$
    – Maxime Ramzi
    Jul 1, 2019 at 17:12
  • $\begingroup$ you are right. Isomorphic over set is indeed easy and i'm not really looking at equivalence, so i am interested in the case of "abstract categories". I m thinking of maybe a canonical construction of the monad out of the algebra category $\endgroup$
    – Lilolance
    Jul 1, 2019 at 19:02
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    $\begingroup$ Then the answer to 'are they isomorphic as endofunctors' is no, and it's in the theory of Morita equivalence that you will find relevant things about your other questions. For instance, the rings $R$ and $M_n(R)$ are Morita equivalent, and of course the category of $S$-modules for any ring $S$ is the category of algebras over a monad associated to $S$, and one may recover the ring $S$ from the monad associated to it (but not from the category of algebras, as non-isomorphic but Morita equivalent rings illustrate) $\endgroup$
    – Maxime Ramzi
    Jul 1, 2019 at 19:22
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    $\begingroup$ See e.g. math.stackexchange.com/questions/2801838/… $\endgroup$
    – Maxime Ramzi
    Jul 1, 2019 at 19:25

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