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

TL;DR

This paper extends framed combinatorial topology by incorporating labels in -categories, creating a new model that facilitates the study of higher categories and generalized tangles through combinatorial data.

Contribution

It introduces a novel extension of framed combinatorial topology with -category labels and constructs -categories classifying meshes and trusses, enabling advanced models of higher categories.

Findings

Develops an extended FCT framework with -category labels

Constructs -categories classifying combinatorial structures

Lays groundwork for models of higher categories and tangles

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.