Diagrammatic notation has become a ubiquitous computational tool; early examples include Penrose’s graphical notation for tensor calculus, Feynman’s diagrams for perturbative quantum field theory, and Cvitanovic’s birdtracks for Lie algebras. Category theory provides a robust framework in which to understand the nature of such diagrams, and Joyal and Street formalized this framework by introducing string diagrams, governed by the syntax of monoidal 1-categories. The notion of “manifold diagrams” generalizes string diagrams to higher dimensions, and can be interpreted in higher-categorical terms by a process of geometric dualization. The closely related notion of “tame tangles” describes a well-behaved class of embedded manifolds that can likewise be interpreted categorically. In this paper we formally introduce the notions of manifold diagrams and of tame tangles, and show that they admit a combinatorial classification, by using results from the toolbox of framed combinatorial topology. We then study the stability of tame tangles under perturbation; the local forms of perturbation stable tame tangles provide combinatorial models of differential singularities. As an illustration we describe various such combinatorial singularities in low dimensions. We conclude by observing that all smooth 4-manifolds can be presented as tame tangles, and conjecture that the same is true for smooth manifolds of any dimension.

Idea

Roughly (and incompletely), the paper contains:

Sec. 1: A concise recollection of trusses, meshes and tame stratifications (Ch. 2, Ch. 4, Ch. 5, and App. B in the FCT book)

Sec. 2: A topological definition of manifold diagrams by a simple generalization of conical stratifications to framed spaces; a discussion of PL and smooth properties of manifold diagrams; a fully combinatorial classification of framed homeomorphisms classes of manifold diagrams.

Sec. 3: A topological definition of tangles; a discussion of their relation to manifold diagrams; a combinatorial classification; work towards using tangles as a foundation for singularity theory (and a empiric discussion of the relation to classical singularity theory); conjectures about how tangles “combinatorialize” smooth strctures.

It does not contain:

Actual “Geometric” Higher Category Theory (i.e. higher category theory based on framed combinatorial-topological structures)