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I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the category of $\mathbb{N}$-Modules also live in a category of categories, namely a category of categories of modules, $\mathcal{C}_{Mod}$. Since the monads compose, I am guessing that the categories of modules themselves compose in the category $\mathcal{C}_{Mod}$. We should, then, expect some kind of composition in this category. A first guess is that of a tensor product, so my first question is just whether this is true: monad composition induces tensor product. What is the standard name for the inducement where a composition of monads induces a tensor product of modules?

Edit: I think this is a monoidal functor between the category of modules and the category Mon_set, of monads on SET.

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    $\begingroup$ In the question you linked, you learned that the composition not simply IS a monad but can be MADE into a monad using additional structure called a distributive law - With possibly more than one possibility to chose from for a given pair of monads! So there is no reason to expect a product of module categories: For a given pair of module categories you'd have to specify additional structure "corresponding" to distributive laws. You'd then expect there to be again more than one possibility for such a datum, preventing you from canonically defining a tensor product. $\endgroup$
    – Gerrit Begher
    Commented Jun 3, 2019 at 4:33
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    $\begingroup$ There might be another possibility: The monads you want to compose are finitary monads. Such a monad $M:\mathrm{Set}\to\mathrm{Set}$ is already defined by its restriction $M_{\mathrm{fin}}:\mathrm{FiniteSets}\to\mathrm{Set}$. You could then look at the monad defined by $M_{\mathbb C}\circ(M_{\mathbb N})_{\mathrm{fin}}$ or $M_{\mathbb N}\circ(M_{\mathbb C})_{\mathrm{fin}}$. Maybe these do are what you are looking for? --- arxiv.org/pdf/0704.2030.pdf $\endgroup$
    – Gerrit Begher
    Commented Jun 3, 2019 at 4:42

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