Hi, This can be regarded as a special case of the generalised nerve theorem of Mark Weber [2007, Familial 2-functors and parametric right adjoints], which states that for a suitable subcategory A ("arities") of C, the Eilenberg-Moore category for T embeds into the presheaf category on the full subcategory of the Kleisli category spanned by A. Modulo Weber's requirement that A be small, C is always suitable. Best wishes, -- Zhen Lin On 21 August 2012 17:34, <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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