Welcome to MathOverflow

Question

5

1

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

flag	asked Mar 2 at 20:14 Ben 191●5

Shouldn't such a functor also interchange Zinb and Leib? – Dan Petersen Mar 2 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 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 Mar 2 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 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 Mar 3 at 0:55

show 1 more comment

Is the operadic butterfly symmetric?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.