Yoneda Lemma when there is a monad
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Chris,
C^T embeds in [(C_T)^op,Set]; i.e. any category of algebras embeds in the presheaf category of the Kleisli category.
This part is morally clear, I think, via the notion of density. A functor F: A --> B is called dense if the induced functor Hom(F, -): B --> [A^op, Set] is full and faithful. An equivalent condition (stated loosely) is that every object of B is a colimit of objects of the form F(a) (with a in A) in a canonical way. (See e.g. Categories for the Working Mathematician.) Now take a monad T on a category C. Every T-algebra is canonically a coequalizer of free T-algebras. So every object of C^T is canonically a colimit of objects of the subcategory C_T. As long as "canonically" means what I assume it does (and I haven't checked the details), this tells us that the inclusion C_T --> C^T is dense. Hence the induced functor C^T ---> [(C_T)^op, Set] is full and faithful - which I guess is what you mean by an embedding. All the best, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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.
The square appears in Section 9 of [FEJ Linton, An Outline of Functorial Semantics, Lecture Notes in Math 80]; see page 32 of [Reprints in Theory and Applications of Categories 18 (2008)1--303]. At a quick look, it doesn't seem to say that the square is a pullback at that point. ==Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
-
info＠christophertownsend.org -
Mark Weber -
Ross Street -
Tom Leinster -
Vaughan Pratt -
Zhen Lin Low