Re: Yoneda Lemma when there is a monad

[Ross Street <ross.street@mq.edu.au>]

Dear Christopher

On 21/08/2012, at 7:34 PM, info@christophertownsend.org wrote:

> The usual Yoneda lemma is recovered by taking the trivial monad. The
> Lemma gives a generalised Yoneda embedding: C^T embeds in
> [(C_T)^op,Set]; i.e. any category of algebras embeds in the presheaf
> category of the Kleisli category. I wasn't aware of this quite trivial
> result and was hoping for some guidance as to where it is covered
> already in the literature?

Indeed, the category C^T of Eilenberg-Moore algebras is the pullback of
the Yoneda embedding y : C --> [C^op,Set] and the functor
[K^op,1] : [(C_T)^op,Set] --> [C^op,Set] which restricts along K.

The pullback appears with proof on page 166 of 

[The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149--168] 

and is proved in the setting of Yoneda structures in

[(with R.F.C. Walters) Yoneda structures on 2-categories, J. Algebra 50 (1978) 350--379].

I attribute the result to 

[FEJ Linton, Relative functorial semantics: adjointness results, Lecture Notes in Math 99 (1969) 166--177]

but cannot remember whether the pullback is explicitly there.

Best wishes,
Ross

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