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? Thank you, David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]