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Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\mathcal{Gerst}$. Algebras over that operad have a commutative product, degree $-1$ Lie bracket, and satisfy a compatibility condition analogues to the Poisson identity.

As was defined by Kontsevich and Manin, taking homology of the genus zero Deligne-Mumford operad $$\left\{\overline{\mathcal{M}}_{0,d+1}\right\}_d$$ defines a non-unital, genus zero, tree-level reduction of a CohFT (or a Frobenius manifolds equivalently).

Finally, consider the real locus of the DM-spaces. This spaces are known to $K(\pi,1)$ and their fundamental group is called the pure cacti group. It plays a rule analogous to that of the braid group in representation theory is has some interesting applications, e.g., for coboundary categories. The sequence $$\left\{\overline{\mathcal{M}}_{0,d+1}(\mathbb{R})\right\}_d$$ for all $d \geq 1$ defines the Mosaic operad of Devadoss, which is made by ''choping and patching " copies of Tamari-Stasheff associahedra.

whose homology was famously computed by P. Etingof, A. Henrquies, J. Kamnitzer, and E. Rains in: ''The cohomology ring of the real locus of the moduli space of stable curves". They concludes that algebras over this operad are ''2-Gerstenhaber" which means they have a commutative product and a ''2-Lie" bracket.

Question.

  1. What is the algebraic relation between 2-Gerstenhaber algebras and Gerstenhaber algebras?
  2. What is the relation between the Gerstenhaber operad and the Mosaic operad in geometric terms? (i.e. can I define a map that sends cycles to cycles in such a way that I see the relation in 1?)
  3. I believe that there since the mosaic operad can be seen as the (homotopy) fixed points loci of the DM-operad under involution, there should be some kind of a short exact sequence of operads involving orbit spectra and Tate cohomology (like the one that appeared in Westerland) which would explain the relation ... does anybody know if there is any description of such an operad and algebras over it?
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Well, for your question 1 you presumably may ask yourself first about a relationship between (shifted) Lie algebras and Lie 2-algebras. Lie 2-algebras of Hanlon and Wachs can be viewed as $L_\infty$-algebras where all operations except for the ternary one vanish, and this is the only relationship to Lie algebras that I know. I suppose that from this you can extract a result relating 2-Gerstenhaber algebras to a very special kind of homotopy Poisson algebras. I do not think there is more than that.

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  • $\begingroup$ @Dotsenko Thanks, that's helpful (and makes a lot of sense). Is there a chance you know of some reference to question 2 or 3? I'm really trying to tie the mosaic operad to some form of CohFT... $\endgroup$
    – Nati
    Sep 25, 2017 at 17:54
  • $\begingroup$ @Nati you might also want to look at the paper arxiv.org/abs/1905.04499 which appeared after I wrote my answer; you might find some connections via that work $\endgroup$
    – Vladimir Dotsenko
    Feb 3, 2023 at 17:51

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