Posts by Year

Principles of higher-dimensional logic

first published

We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic.

Towards computable manifolds

first published

In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles.

The $D_4$ dualizability law

first published

This note gives a brief visual guide to how one can understand the classical $$D_4$$ singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle).

Github discussions and writing mathematics (deprecated)

first published

Github markdown does not allow for easily writing mathematical formulaes. We discuss some work-arounds, pairable with the comment plugin giscus.

The categorical Pontryagin-Thom construction

first published

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

first published

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

first published

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.

first published

We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; 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 introduce the paradigm of isotopy.

Manifold diagrams: a geometric approach

first published

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.

2021

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

Enriched cohesive infinity toposes

first published

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.

first published

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.