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I realize this question is not research level, but I think this might be just above the abilities of the folks at Mathematics stack. I want a general prescription, or as many as possible, for constructing a monoidal category like FD-Hilb or Hilb given only a monad. I was thinking of using the endo-functor composition as the monoidal product but didn't really get anywhere. We know that the monad axioms are much like the monoidal axioms, though a category like FD-Hilb has many more axioms than can be identified one to one with the monad axioms. Has anyone seen attempts at constructing a basic symmetric monoidal categories from some monad? In particular, I want the monad to be an adjunction of forgetful and free functors, I don't care what the base categories are. Moreover, I want a general prescription: given any free-forgetful adjoint here's how to construct a monoidal category. Help will be most appreciated.

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All categories of monads are defined from monads associated to adjunctions where the right adjoint is a sort of forgetful functor. Conversely, given an adjunction there are criteria to decide wether the source of the right adjoint is the category of algebras over the monad associated to the adjunction. All this is standard category theory, see Borceux's monographs. When the (candidate to) category of algebras is monoidal, Hopf monads are considered. See for instance the work of Moerdijk on this topic. – Fernando Muro Aug 15 '12 at 15:16
Thanks for the suggestions :) – Ben Sprott Aug 15 '12 at 16:25

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