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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