This book lays foundations of a theory of framed combinatorial topology. The theory synthesizes ideas from classical combinatorial topology with a combinatorial notion of framing. The synthesis has interesting consequences, including the following. Firstly, the presence of combinatorial framings allows to overcome fundamental obstruction to computability questions in classical combinatorial topology. Secondly, the framed setting provides a faithful bridge of combinatorics and topology allowing, for instance, for a proof of a `framed local Hauptvermutung'. The scope of the book is in some sense limited: eventually, in a yet bigger picture, framed combinatorial topology aims to provide a combinatorial, and, importantly, computable, foundation to structured algebraic topology and geometric higher category theory; this, for instance, merges ideas from manifold singularity theory, stratified Morse theory, and combinatorial higher category theory into a single unified theory.
While higher groupoids have a natural model in spaces, higher categories have no such well-accepted model. This makes the question of correctness of a given definition of higher categories difficult to answer. We argue that the question has a simple answer “locally”, namely, categories are locally modelled on so-called manifold diagrams. The idea of locally modelling higher categories by manifold diagrams (most prominently in the 3-dimensional case of Gray-categories) is not new and has been proposed by multiple authors. However, the niceness of this manifold-based perspective on higher categories has in the past been somewhat obfuscated by the complexity of manifold geometry in higher dimensions. My thesis discusses a fully algebraic (and - in some sense - canonical) formulation of manifold diagrams, and thereby resolves the long-standing problem of finding a generalization of string diagrams to higher dimensions. Based on this notion of manifold diagrams, we further define “Associative n-categories”, which generalize Gray-categories to higher dimensions.
Some remarks: The first (roughly) 100 pages are a concise summary of all ideas, and should be an easy and self-contained read for most readers familier with basic ordinary category theory. The remaining 400 pages are mostly rather detailed and concrete computations, which explore the combinatorial properties of the theory.
Unpublished and expository writing
The precursor combinatorics to my thesis, modelling higher “sesqui-categories” and not full higher categories.
Expository notes, attempting to give a more modern treatment of basic Algebraic Topology, and then discuss its abstraction in model categories and quasicategories.
Very basic enriched category theory, essentially adding details and computations to Kelly’s book.