**Background.** As we know from Fred Cohen's [Thesis](https://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf), 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](https://arxiv.org/abs/hep-th/9402147), 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](https://arxiv.org/abs/math/9807010) 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"](http://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p03-s.pdf). 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](https://arxiv.org/abs/math/0605080)) which would explain the relation ... does anybody know if there is any description of such an operad and algebras over it?



What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
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. 


