# 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
{"ft_lang_label":"__label__en","ft_lang_prob":0.5964185,"math_prob":0.9924886,"size":951,"snap":"2019-43-2019-47","text_gpt3_token_len":280,"char_repetition_ratio":0.156283,"word_repetition_ratio":0.0,"special_character_ratio":0.28286016,"punctuation_ratio":0.13414635,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9962951,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-14T08:23:45Z\",\"WARC-Record-ID\":\"<urn:uuid:d9e58a7f-6232-4851-9ad9-af40c84c66fa>\",\"Content-Length\":\"119180\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a8e59c15-11cd-465b-ab14-80fe436c3144>\",\"WARC-Concurrent-To\":\"<urn:uuid:2575e607-9ded-453c-bc34-94efe2dd2b79>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/123442/is-the-operadic-butterfly-symmetric\",\"WARC-Payload-Digest\":\"sha1:Y5GZBDXBJAWCTOPBD74BVCMOS4BBO5RJ\",\"WARC-Block-Digest\":\"sha1:N6YU4VGA3FLH2TWXOO4FKHKPVXXCCIGT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496668334.27_warc_CC-MAIN-20191114081021-20191114105021-00531.warc.gz\"}"}