Let be the moduli space of stable algebraic curves of genus with marked points. With the operations that relate the different moduli spaces identifying marked points, the family is a modular operad of projective smooth Deligne-Mumford stacks . In this paper, we prove that the modular operad of singular chains is formal, so it is weakly equivalent to the modular operad of its homology . As a consequence, the up-to-homotopy algebras of these two operads are the same. To obtain this result, we prove a formality theorem for operads analogous to the Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.
V. Navarro. P. Pascual. A. Roig. F. Guillén Santos. "Moduli spaces and formal operads." Duke Math. J. 129 (2) 291 - 335, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12924-6