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We propose a generalization of the concept of a Ribbon graph suitable toprovide combinatorial models for marked surfaces equipped with a G-structure.Our main insight is that the necessary combinatorics is neatly captured in theconcept of a crossed simplicial group as introduced, independently, byKrasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic categoryleads to Ribbon graphs while other crossed simplicial groups naturally yielddifferent notions of structured graphs which model unoriented, N-spin, framed,etc, surfaces. Our main result is that structured graphs provide orbicelldecompositions of the respective G-structured moduli spaces. As an application,we show how, building on our theory of 2-Segal spaces, the resulting theory canbe used to construct categorified state sum invariants of G-structuredsurfaces.

Crossed simplicial groups and structured surfaces