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A strong monad over a monoidal category is a monad in the bicategory of -actions.
Here we regard as equipped with the canonical -action on itself.
If we write for the one-object bicategory obtained by delooping once, we have
where on the right we have the -category of lax 2-functors from to Cat, lax natural transformations of and modifications.
The category defines a canonical functor .
The strong monad, being a monad in this lax functor bicategory is given by
By the general logic of -transformations the components of are themselves a certain functor.
Then the usual diagrams that specify a strong monad
Usually strong monads are described explicitly in terms of the components of the above structure. The above repackaging of that definition is due to John Baez
Strong monads are important in Moggi’s theory of notions of computation (see monad (in computer science)):
- Eugenio Moggi. Notions Of Computation And Monads. Information And Computation. 1991;93:55–92.
- Eugenio Moggi. Computational Lambda-Calculus and Monads. Proceedings of the Fourth Annual Symposium on Logic in Computer Science. 1989. p. 14–23.