The goal of this article is to introduce basic concepts in the emerging area of geometric higher category theory. The exposition aims to be brief and informative but mathematically as self-contained as possible (some familiarity with higher category theory is presumed). Several small exercises (with solutions) have been included, and plenty of open questions as well as future research directions in the area are discussed.
In this expository note, we give a brief overview of some recently developed ideas in the study of framed combinatorial spaces, and its applications in the definition of manifold diagrams, tame tangles, and the combinatorial study of higher Morse theory.
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.
fixed a few (of many) inaccuracies, added and improved material (most importantly, added “bundle” generalizations of some central results in Ch. 4 and 5)
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial topology with a new combinatorial approach to framings. The resulting notion of framed combinatorial spaces has unexpectedly good behavior when compared to classical, nonframed combinatorial notions of space. In discussing this behavior and its contrast with that of classical structures, we emphasize two broad themes, computability in combinatorial topology and combinatorializability of topological phenomena. The first theme of computability concerns whether certain combinatorial structures can be algorithmically recognized and classified. The second theme of combinatorializability concerns whether certain topological structures can be faithfully represented by a discrete structure. Combining these themes, we will find that in the context of framed combinatorial topology we can overcome a set of fundamental classical obstructions to the computable combinatorial representation of topological phenomena.
We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to “directed spaces”. Directed spaces will be locally modelled on manifold diagrams, which are stratifications of the $n$-cube such that strata are transversal to the flag foliation of the n-cube. The first part of this thesis develops a combinatorial language for manifold diagrams called singular n-cubes. In the second part we apply this language to build our notions of higher categories.
Singular n-cubes can be thought of as “flag-foliation-compatible” stratifications of the n-cube, such that strata are “stable” under projections from the $(k + 1)$- to the $k$-cube, together with a functorial assignment of data to strata. The definition of singular n-cubes is inductive, with $(n + 1)$-cubes being defined as combinatorial bundles of n-cubes over the (stratified) interval. The combinatorial structure of singular n-cubes can be naturally organised into two categories: $\mathbf{SI}//^n_C$, whose morphisms are bundles themselves, and $\mathbf{Cube}^n_C$, whose morphisms are inductively defined as base changes of bundles. The former category is used for the inductive construction of singular n-cubes. The latter category describes the following interactions of these cubes. There is a subcategory of “open” base changes, which topologically correspond to open maps of bundles. We show this subcategory admits an (epi,mono) factorisation system. Monomorphism will be called embeddings and describe how cubes can be embedded in one another such that strata are preserved. Epimorphisms will be called collapses and describe how strata can be can be refined. Two cubes are equivalent if there is a cube that they both refine. We prove that each “equivalence class” (that is, the connected component of the subcategory generated by epimorphisms) has a terminal object, called the collapse normal form. Geometrically speaking the existence of collapse normal forms translates into saying that any combinatorially represented manifold diagram has a unique coarsest stratification, making the equality relation between manifold diagrams decidable and computer implementable.
As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. In particular, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. The first depends on a theory of sets, while the latter two do not, making them a step towards a framework for working with general higher categories in a foundation- independent way. All three notions will have strict units and associators. The only “weak” coherences which are present will be called homotopies. We propose that this is the right conceptual categorisation of coherence data: homotopies are essential coherences, while all other coherences can be uniformly derived from them. As evidence to this claim we define presented weak n-categories, and develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This motivates the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.
Executive summary
This thesis works towards a combinatorial notion of manifold diagrams by introducing the novel combinatorial structure of singular \(n\)-cubes labeled in \(\mathsf{C}\) (these are now, in more general form, called “\(n\)-trusses [labeled] in \(\mathsf{C}\)”). Based on a combinatorial definition of manifold diagrams (N.B: no real attempt is made at formalizing manifold diagrams in geometric terms), the thesis goes on to define combinatorial models of both free and non-free higher categories. It also sketches a path towards the categorical study of singularities and the Pontryagin-Thom construction. However, it certainly stops short of giving a full account of these ideas (in particular, it does not compare either notion to other models of higher categories).
