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.

Arboreal singularities

“…(2) the universal planar tree over the (n−2)-dimensional associahedron K n−2 ( [13,21,24]), (3) in geometric realizations of Waldhausen's S-construction in K-theory ( [8,9,15]). …”

Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

“…The nature of the above principle is similar to the cobordism hypothesis [11] as they both describe objects of geometric nature by purely categorial means, see again [8] where the relation to the structured cobordism hypothesis in dimension 2 was explicitly formulated, cf. also Remark 4.3.7 of [8].…”

“…Several steps in this direction have already been made in [8] where various generalizations of dihedral, quaternionic and other generalizations of cyclic structures based on crossed simplicial groups of Krasauskas [25] and Fiedorowicz-Loday [10] were studied, and their relation to structured surfaces was clarified.…”

Homotopy of planar Lie group equivariant presheaves

“…Detailed accounts of the properties of these objects can be found in [9]. Here we will recall the necessary notions that we need for our intended application.…”

“…Planar crossed simplicial groups are an interesting thing to study as their geometric realisations are planar Lie groups, which can be used to add extra structure to surfaces as done in [9]. Planar Lie groups have a full classification, and this classification matches up exactly with the planar crossed simplicial groups.…”