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).
Meta-comments
Upon observing an apparant inductive structure in some (heuristic, but not formalized) idea of what manifold diagrams should be, I wrote out the combinatorial theory of “singular cubes”. The theory of these combinatorial structures is excitingly rich and a lot of things just work nicely, which led to the thesis “writing itself” and resulted in a rather long document (one may argue that this is also due to me writing out all calculations and verifications in quite a lot of (…way too much) detail). The theory was subsequently generalized in absorbed into the theory of “trusses” developed with Christopher Douglas, based on the realization that, really, “framed combinatorial-topological models” should use the duals of the structures considered in my thesis. See the book for more details, and a paper that makes the underlying intuition about manifold diagrams formal.
Errata
There are (at least) a couple of claims that are wrong. I’ll hope to eventually write out a list of issues here.
