Re: monad: (k-Set \downarrow -): Set -->Set

In the case where k=omega, T is the well-known finite multiset monad, which associates to each S the free commutative monoid generated by S (whose elements are also known as finite multisets in S).

For other k, I would call this the "monad of multisets of size less than k". I think this works for any infinite small cardinal, not just regular ones.

You really do need the regularity. Otherwise removing brackets from a k-small multiset of k-small multisets might yield something bigger than a k-small multiset, and then one cannot define a multiplication for the monad. Richard