# Is the operadic butterfly symmetric?

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: $(Dend,Dias) \leftrightarrow (Comm,Lie)$.

• 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
• 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

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)?