The operad structure of admissible G-covers

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible G-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a category. This construction interpolates in a sense between "framed" and "colored" versions of operads; we hope that it will be of independent interest. An algebra over the cohomology of this operad is the same thing as a G-equivariant CohFT, as defined by Jarvis, Kaufmann and Kimura. We prove that the (orbifold) Gromov-Witten invariants of global quotients [X/G] give examples of G-CohFTs.