# Notes

## The categorical Pontryagin-Thom construction

updated

The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether.

## From trusses to weak computads to combinatorial manifold diagrams

updated

This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to weak computads, and how this can be used to understand manifold diagrams in purely combinatorial terms.

## Tame tangles, singularities, and Morse theory

updated

Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory.

## Paradigms of higher category theory

updated

Most models for higher categories are based on the paradigm of contractibility; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. To remedy this, we discuss the paradigm of homotopicity.

## Manifold diagrams: a geometric approach

updated

We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells.

## The class of computadic cells

updated

We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology.

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology and higher algebra.

## Enriched cohesive infinity toposes

updated

I gave a talk in the Advanced Topology Class at Oxford about “Generalized Differential Cohomology” based on this book. Part of the talk required setting up an “enriched cohesive $$\infty$$-topos”. Here’s an outline of what that object could be and what it does.

## Factorization homology

updated

I gave a talk at CUNY about factorization homology, and have written up some notes which I’m posting here.

## Github discussions and writing mathematics

updated

I switched from MathJax to Katex, and from giscus to a fork of Commento++ which fixed some bugs in the latter and added Katex support. I intend on making that fork public soon and write about it here.