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

[Mark.Weber@pps.jussieu.fr]

Dear David

I'm not sure whether your specific construction has been looked at or
named, but it is a "submonad" of something very well known.

Let's work with 2 universes of sets, your "Set" being the topos of small
sets, and SET being a topos of sets large enough to include {arrows of
Set} as an object. For those not so comfortable doing this, fix two
regular cardinals far enough part, read "Set" as "the category of sets of
cardinality less than the smaller cardinal", and "SET" as "the category of
sets of cardinality less than the bigger one", and far enough apart means
Set is a category internal to SET.

Now the 2-category CAT (of category objects in SET) contains Set as an
object so one may form the slice 2-category

CAT/Set

in the strictest sense (1-cells being triangles commuting on the nose).
The big brother of your monad is a 2-monad on this 2-category. The
endofunctor part does the following:

S:A-->Set   |-->  Set \downarrow S --> Set

This is the underlying monad of what could be called the "fibration"
2-monad. That is applying to a functor produces the free split fibration
on what you started with. This construction works at the following
generality: replace CAT by any 2-category with comma objects and Set by
any object therein, and the first paper to see fibrations as algebras of a
monad in this way was

R. Street "Fibrations and yoneda's lemma in a 2-category" SLNM 420 1974

The relation between your monad and this one is that there's a canonical
inclusion

Set --> CAT/Set

which regards any Set S as a functor S:1-->Set, and this functor is the
1-cell data for a monad morphism (in the sense of Street: "Formal theory
of monads") from your monad to the monad I described.

With best regards

Mark Weber

> 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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