  This is preprint HAL-00429963 (2009). I describe a combinatorial construction
of the cohomology classes in compactified moduli spaces of curves
$\widehat{Z}_{I}\in H^{*}(\bar{\mathcal{M}}_{g,n})$ starting from the following
data: an odd derivation $I$, whose square is non-zero in general, $I^{2}\neq
0$, acting on a $\mathbb{Z}/2\mathbb{Z}$-graded associative algebra with odd
scalar product. The constructed cocycles were first described in the theorem 2
in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix
integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:0912.5484 ,
preprint HAL-00102085 (09/2006). By the theorem 3 from loc.cit. the family of
the cohomology classes obtained in the case of the algebra $Q(N)$ and the
derivation $I=\left[\Lambda,\cdot\right]$ coincided with the generating
function of products of $\psi-$classes. This was the first nontrivial
computation of categorical Gromov-Witten invariants of higher genus. The result
matched with the mirror symmetry prediction, i.e. with the classical
(non-categorical) Gromov-Witten descendent invariants of a point for all genus.
As a byproduct of that computation a new combinatorial formula for products of
$\psi$-classes $\psi_{i}=c_{1}(T_{p_{i}}^{*})$ in the cohomology
$H^{*}(\bar{\mathcal{M}}_{g,n})$ is written out.
