strong monad


Higher algebra

2-Category theory

Strong monads


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



For VV a monoidal category a strong monad over VV is a monad

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


If we write BV\mathbf{B}V for the one-object bicategory obtained by delooping VV once, we have

V-ActLax2Funct(BV,Cat), V\text{-}Act \simeq Lax2Funct(\mathbf{B}V, Cat) \,,

where on the right we have the 22-category of lax 2-functors from BV\mathbf{B}V to Cat, lax natural transformations of and modifications.

The category VV defines a canonical functor V^:BVCat\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:V^V^T : \hat V \to \hat V;

  • modifications

    • unit: η:Id VT\eta : Id_V \Rightarrow T

    • product: μ:TTT\mu : T \circ T \Rightarrow T

  • satisfying the usual uniticity and associativity constraints.

By the general logic of (2,1)(2,1)-transformations the components of TT are themselves a certain functor.

Then the usual diagrams that specify a strong monad

  • unitalness and functoriality of the component functor of TT;

  • naturalness of unit and product modifications.


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

Revised on March 11, 2014 02:53:37 by Urs Schreiber (