Re: Yoneda Lemma when there is a monad

[Vaughan Pratt <pratt@cs.stanford.edu>]

If this has an answer I'd like to know.  In my paper

Pratt, V.R. ``Communes via Yoneda, from an Elementary Perspective'',
Fundamenta Informaticae 103, 203-218, DOI 10.3233/FI-2010-325,
IOS Press, 2010

which among other things generalizes it in section 1.5 to the Chu
setting, I treated it as part of the uncitable Yoneda folklore.

Why uncitable?  Well, one of the many Norbert Weiner stories is that he
was challenged in class on some point.  He stared at the board for a
bit, wandered outside, came back twenty minutes later and said "It's
obvious."

The Yoneda lemma is like that.

Vaughan Pratt


On 8/21/2012 2:34 AM, 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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