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.)