Re: Codensity and the ultrafilter monad

[Michel Hebert <mhebert@aucegypt.edu>]

Hi Tom,

This appears as exercise 3.2.12(e) in Manes' book (Algebraic Theories). The
references there are Lawvere's thesis and Linton's 1966, but I don't know
if this part of the exercise is solved or mentioned explicitly there.
Best regards,
Michel

On Fri, Jun 10, 2011 at 12:25 AM, Tom Leinster
<Tom.Leinster@glasgow.ac.uk>wrote:

> Dear all,
>
> Any functor from a small category A to a complete category E induces a
> contravariant adjunction between E and Set^A.  This in turn induces a monad
> on E, the "codensity monad" of the functor.
>
> (The construction of the adjunction is better known in its dual form,
> starting with a functor from a small category to a COcomplete category. For
> example, the usual functor from Delta into Top induces the usual adjunction
> between topological spaces and simplicial sets.)
>
> The codensity monad of the inclusion FinSet --> Set is the ultrafilter
> monad.  This seems a rather basic fact, but I've been unable to find it in
> the literature.  I'd be grateful if someone could tell me a reference.
>
> (I'm aware of the 1987 paper by Reinhard Börger giving a different but
> related characterization of the ultrafilter monad.)
>
> Thanks,
> Tom

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