On Fri, 19 Jun 2009, 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.
I don't think this construction works at the level of sets rather than categories. The problem is that k-Set is a category, not a set, so T(S) also has a category structure, and you can't simply "forget" this. If you do, then you have the problem "*which* singleton set?" for the unit (i.e., which singleton set do you choose as the domain of the functions 1 --> S which you identify with elements of S?), and whichever choice you make you are going to run into problems verifying the monad identities.
(There is a similar monad on Cat, where we replace k-Set with k-Cat.)
This is correct, and it's well-known: it is the monad which freely adjoins k-small coproducts to a category. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]