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

  • $\begingroup$ Shouldn't such a functor also interchange Zinb and Leib? $\endgroup$ Mar 2, 2013 at 20:21
  • $\begingroup$ Do you also want your horizontal symmetry to extend to the big diagram on page 3 of arxiv.org/pdf/math/0409183v1.pdf? $\endgroup$ Mar 2, 2013 at 20:48
  • $\begingroup$ Dan: An example which does that would be interesting too. André: It is reasonable to ask that $F$ swaps $\chi$ and $Vect$. $\endgroup$
    – Ben Cooper
    Mar 2, 2013 at 23:21
  • 2
    $\begingroup$ 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? $\endgroup$ Mar 2, 2013 at 23:28
  • $\begingroup$ 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. $\endgroup$
    – Ben Cooper
    Mar 3, 2013 at 0:55

1 Answer 1


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)?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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