Dear Christopher Your lemma is a combination of 2 results in the literature. The first is Proposition(13) of 1. R. Street and B. Walters, Yoneda structures on 2-categories, Journal of Algebra 50:350-379, 1978 which says that for a functor J:B-->C between locally small categories, the functor PB(C(J,1),1):PB-->PC corresponds to taking right kan extension along J -- here PB=[B^op,Set], C(J,1):C-->PB sends (b,c) to the hom set C(Jb,c). Your lemma is the case J=K (the comparison functor), and one uses the result that K is dense, due to Fred Linton 2. F. Linton, Relative functorial semantics: adjointness results, LNM 99:384-418, 1969 to simplify the right kan extension to the F(A,a) of your lemma. Often there are nice subcategories of C_T which are also dense in C^T, and this would give other variants of your lemma. For more on this kind of situation see http://arxiv.org/abs/1101.3064 With best regards, Mark Weber On 21/08/2012, at 7:34 PM, info@christophertownsend.org wrote:
Hi
Does anyone have any references for the following generalisation of the Yoneda lemma:
Lemma: If (T,i,m) is a monad on a locally small category C then for any functor F:(C_T)^op->Set, contravariant from the Kleisli category to Set and for any T algebra (A,a)
Nat[C^T(K_,(A,a)),F]=F(A,a)
where K is the usual comparison functor from the Kleisli category, C_T, to the category of algebras of T, C^T. F(A,a) means the subset of F(A) consisting of elements x such that F(m_A)x=F(Ta)x. (And Nat[_] means the set of natural transformations.) //
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?
Thanks, Christopher
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