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

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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


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