Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.

$$\begin{array}{ccccc} & Dend & & & & Dias & \newline \nearrow & &\searrow & &\nearrow & &\searrow\newline Zinb & & &Ass & & & \quad Leib \newline \searrow & &\nearrow & &\searrow & &\nearrow\newline & Comm & & & & Lie & \newline \end{array}$$

Here is a paper by Loday in which some discussion can be found. As the reference explains, the Koszul duality functor, $\mathcal{O}\mapsto \mathcal{O}^!$, gives the above diagram symmetry about the vertical axis.

Is the butterfly also symmetric about the horizontal axis?

In other words, does there exist a functor $F : Operad(C) \to Operad(C)$, where $C$ is vector spaces or chain complexes, such that $F\circ F \simeq 1$, $F$ fixes the operads: $Zinb$, $Ass$ and $Leib$, and exchanges the pairs: $(Dend,Dias) \leftrightarrow (Comm,Lie)$.

share|improve this question
    
Shouldn't such a functor also interchange Zinb and Leib? –  Dan Petersen Mar 2 '13 at 20:21
    
Do you also want your horizontal symmetry to extend to the big diagram on page 3 of arxiv.org/pdf/math/0409183v1.pdf? –  André Henriques Mar 2 '13 at 20:48
    
Dan: An example which does that would be interesting too. André: It is reasonable to ask that $F$ swaps $\chi$ and $Vect$. –  Ben Cooper Mar 2 '13 at 23:21
2  
It looks difficult for any kind of reasonable construction to exchange $\chi$ and $Vect$: The operad $Vect$ is completely trivial (zero generators), while $\chi$ is a mess (actually, there are two versions of it -- which one do you want??). If you think that $\chi$ can come out of $Vect$ by some kind of natural construction, then you are also implicitly saying that $\chi$ is a very important operad and that everybody should know about it: are you claiming that? –  André Henriques Mar 2 '13 at 23:28
    
The larger diagram does suggest investigating funny extensions. An $F$ defined using $\chi$ wouldn't increase its importance (any more than Loday's paper). If such a construction exists then probably $Operad(C)$ has lots of messy symmetries. It just seemed like an interesting question. –  Ben Cooper Mar 3 '13 at 0:55

1 Answer 1

up vote 3 down vote accepted

I have never seen such a functor and I doubt that it would exist. For instance, the operad Com is symmetric and one-dimensional in every arity. Whereas the operad Dend is regular (coming from a non-symmetric one) and is generated by two (non-symmetric) generators. I also now no non-trivial functor which preserves Zinb or Leib.

Can I ask why you are looking for such a functor (despite the symmetric shape of the operadic butterfly)?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.