2-categorical monad functor lifting

2-categorical monad functor lifting

[Steve Vickers]

As we know, a monad functor between two monads (possibly on different
categories) can be lifted to a functor between the algebra categories.

Fauser and I needed a corresponding 2-categorical result: a 2-monad
2-functor (of course, the natural transformation involved needs to be
2-natural) between two 2-monads (possibly on different 2-categories) can
be lifted to a 2-functor between the lax algebra 2-categories. Also the
lifting preserves pseudo-ness and strictness of the algebras.

We checked all the equations and it works, but we're not experts on the
2-categorical literature and we wonder if it's already known. Our first
literature searches haven't shown up anything - though even the
1-categorical case (which surely is well known) is elusive.

Has anyone seen this result before?

Steve Vickers.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: 2-categorical monad functor lifting

[Francisco Marmolejo (314)]

Dear Steve,

you may want to check

Coherence for pseudodistributive laws revisited
F. Marmolejo and R.J. Wood
Theory and Applications of Categories, Vol. 20, No. 6, 2008, pp. 74–84.

and the bibliography there-in for the pseudo case (and for pseudomonads). 
The standard reference for the 1-categorical case must surely be

The formal theory of monads
R. Street
J. Pure Appl. Algebra 2 (1972)

F. Marmolejo


On Mon, 26 Mar 2012, Steve Vickers wrote:

> As we know, a monad functor between two monads (possibly on different
> categories) can be lifted to a functor between the algebra categories.
>
> Fauser and I needed a corresponding 2-categorical result: a 2-monad
> 2-functor (of course, the natural transformation involved needs to be
> 2-natural) between two 2-monads (possibly on different 2-categories) can
> be lifted to a 2-functor between the lax algebra 2-categories. Also the
> lifting preserves pseudo-ness and strictness of the algebras.
>
> We checked all the equations and it works, but we're not experts on the
> 2-categorical literature and we wonder if it's already known. Our first
> literature searches haven't shown up anything - though even the
> 1-categorical case (which surely is well known) is elusive.
>
> Has anyone seen this result before?
>
> Steve Vickers.
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]