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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]