nLab
strong monad

Idea

A strong monad over a monoidal category $V$ is a monad in the bicategory of $V$-actions.

Definition

Definition

For $V$ a monoidal category a strong monad over $V$ is a monad

• in the $2$-category $V\text{-}\mathrm{Act}$ of left $V$-actions on categories

• on the object $V$ itself.

Here we regard $V$ as equipped with the canonical $V$-action on itself.

Details

If we write $BV$ for the one-object bicategory obtained by delooping $V$ once, we have

$$V\text{-}\mathrm{Act}\simeq \mathrm{Lax}2\mathrm{Funct}(BV,\mathrm{Cat}),$$V\text{-}Act \simeq Lax2Funct(\mathbf{B}V, Cat) \,,

where on the right we have the $2$-category of lax 2-functors from $BV$ to Cat, lax natural transformations of and modifications.

The category $V$ defines a canonical functor $\hat{V}:BV\to \mathrm{Cat}$.

The strong monad, being a monad in this lax functor bicategory is given by

• a lax natural transformation $T:\hat{V}\to \hat{V}$;

• modifications

  • unit: $\eta :{\mathrm{Id}}_V\Rightarrow T$

  • product: $\mu :T\circ T\Rightarrow T$

• satisfying the usual uniticity and associativity constraints.

By the general logic of $(\mathrm{2,1})$-transformations the components of $T$ are themselves a certain functor.

Then the usual diagrams that specify a strong monad

• unitalness and functoriality of the component functor of $T$;

• naturalness of unit and product modifications.

References

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

• John Baez, The Monads Hurt My Head – But Not Anymore (blog)