Crossed simplicial groups and structured surfaces

Abstract

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

86 pages, v2: revised version

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• On the Classification of Topological Field Theories
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• Triangulated surfaces in triangulated categories
• The A-infinity operad and the moduli space of curves

Cited by in corpus (13)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Relative Calabi-Yau structures
• A1-homotopy invariants of topological Fukaya categories of surfaces
• Topological field theories on open-closed $r$-spin surfaces
• Relative $2$-Segal spaces
• Ginzburg algebras of triangulated surfaces and perverse schobers
• Reflexive homology
• Spherical adjunctions of stable $\infty$-categories and the relative S-construction
• Perverse sheaves on Riemann surfaces as Milnor sheaves
• Structured Topological Field Theories via Crossed Simplicial Groups
• Reflexive homology and involutive Hochschild homology as equivariant Loday constructions
• Limits and colimits of crossed groups
• Categorifying equivariant monoids