Framed Combinatorial Topology with Labels in $\infty$-Categories

Abstract: Framed combinatorial topology is a recent approach to tame geometry which expresses higher-dimensional stratified spaces via tractable combinatorial data. The resulting theory of spaces is well-behaved and computable. In this paper we extend FCT by allowing labelling systems of meshes and trusses in $\infty$-categories, and build an alternative model of FCT by constructing $\infty$-categories that classify meshes and trusses. This will serve as the foundation for future work on models of higher categories based on generalised string diagrams and the study of generalised tangles.