David Spivak wrote:
Dear Categorists,
Does anyone know a name for the monad described below and/or whether it has been studied?
Let k-Set denote the category of k-small sets (for some small regular cardinal k). For a set S, we denote by
T(S)=(k-Set \downarrow {S})
the set whose elements are pairs (K,f), where K is a k-small set and f:K-->S is a function. This construction is functorial in S. I claim that the endo-functor T: Set -->Set is a monad. The identity transformation S-->T(S) is given by "singleton set" and the multiplication transformation TT(S)-->T(S) is given by Grothendieck construction.
(There is a similar monad on Cat, where we replace k-Set with k-Cat.)
Does this monad T have a name? Has it been studied?
I assume you mean to take such pairs (K,f) up to isomorphism, or else, as Peter Johnstone has already pointed out, your construction will not be well-defined. For instance, even the finite sets may form a proper class, depending on your underlying set theory. 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. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]