Re: Yoneda Lemma when there is a monad

[Mark Weber <mark.weber.math@googlemail.com>]

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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