Subject: categories: Monads on finite categories

From: "Tom Leinster" <[email-masked]>
Date: Sat, 25 Mar 2006 21:32:47 -0000 (GMT)

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.)