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

[selinger@mathstat.dal.ca (Peter Selinger)]

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



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