#### Vol. 7, No. 8, 2013

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The operad structure of admissible $G$-covers

### Dan Petersen

Vol. 7 (2013), No. 8, 1953–1975
##### Abstract

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 $\left[X∕G\right]$ give examples of $G$-CohFTs.

##### Keywords
modular operad, operad colored by groupoid, orbifold Gromov–Witten theory, cohomological field theory
##### Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 14H10, 14D21