Re: Monads on finite categories

Re: Monads on finite categories

[Reinhard Boerger]

Hello,

Tom Leinster wrote:

> 1. Are there significant or interesting examples of monads on finite
>    categories?  I want to look beyond monads on posets, a.k.a. closure
>    operators.  (Since a finite category with binary sums or products
>    is necessarily a poset, some of the usual examples of monads reduce
>    to this case.)  I can only think of one class of examples
>    (described below), and I don't know if it's particularly
>    significant.

Idempotent monads correspond to full reflective subcategories; so the only
examples are induced by full reflective subcategories of finite
categories.



Greetings

Reinhard

Monads on finite categories

[Tom Leinster]

Dear All,

1. Are there significant or interesting examples of monads on finite
   categories?  I want to look beyond monads on posets, a.k.a. closure
   operators.  (Since a finite category with binary sums or products
   is necessarily a poset, some of the usual examples of monads reduce
   to this case.)  I can only think of one class of examples
   (described below), and I don't know if it's particularly
   significant.

2. Any monad on a finite category is idempotent.  Is this widely
   known?

Thanks.

Tom

* * *

The class of examples: let A be a finite Cauchy-complete category.
Let M be the 2-element monoid consisting of the identity and an
idempotent, so that [M, A] is the category of idempotents in A.  Then
the diagonal functor A ---> [M, A] has adjoints on both sides.  The
induced monad on A is trivial, but that on [M, A] is not.  (It sends
an idempotent e to 1_a, where a is the object through which e splits.)