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

[David Spivak <dspivak@uoregon.edu>]

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


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