### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### 2-Category theory

2-category theory

## 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 $V\text{-}Act$ 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 $\mathbf{B}V$ for the one-object bicategory obtained by delooping $V$ once, we have

$V\text{-}Act \simeq Lax2Funct(\mathbf{B}V, Cat) \,,$

where on the right we have the $2$-category of lax 2-functors from $\mathbf{B}V$ to Cat, lax natural transformations of and modifications.

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

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: $\eta : Id_V \Rightarrow T$

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

• satisfying the usual uniticity and associativity constraints.

By the general logic of $(2,1)$-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.

## Alternative definition

A more concrete definition is given in:

• Eugenio Moggi. Computational Lambda-Calculus and Monads. Proceedings of the Fourth Annual Symposium on Logic in Computer Science. 1989. p. 14–23.

A strong monad over a category $C$ with finite products is a monad $(T, \eta, \mu)$ together with a natural transformation $t_{A,B}$ from $A \times T\;B$ to $T(A\times B)$ subject to three diagrams.

“Strengthening with 1 is irrelevant”:

(1)$\begin{array}{ccc} 1\times T\, A & \rightarrow & T\, A\\ \\ & t_{1,A}\searrow\phantom{t_{1,A}} & \downarrow\\ \\ & & T(1\times A) \end{array}$

“Consecutive applications of strength commute”:

(2)$\begin{array}{ccccc} (A\times B)\times T\, C & \xrightarrow{t_{A\times B,C}} & T\,((A\times B)\times C)\\ \\ \cong\downarrow\phantom{\cong} & & & \phantom{\cong}\searrow\cong\\ \\ A\times(B\times T\, C) & \xrightarrow[A\times t_{B,C}]{} & A\times T\,(B\times C) & \xrightarrow[t_{A,B\times C}]{} & T(A\times(B\times C)) \end{array}$

“Strength commutes with monad unit and multiplication”:

(3)$\begin{array}{ccccc} A\times B\\ \\ A\times\eta_{B}\downarrow\phantom{A\times\eta_{B}} & \phantom{\eta_{A\times B}}\searrow\eta_{A\times B}\\ \\ A\times T\, B & \xrightarrow{t_{A,B}} & T(A\times B)\\ \\ A\times\mu_{B}\uparrow\phantom{A\times\mu_{B}} & & & \phantom{\mu_{A\times B}}\nwarrow\mu_{A\times B}\\ \\ A\times T^{2}\, B & \xrightarrow{t_{A,TB}} & T(A\times TB) & \xrightarrow{T\: t_{A,B}} & T^{2}(A\times B) \end{array}$

## 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

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.

Revised on October 12, 2015 11:42:35 by Bram Geron (147.188.200.212)