# Geometric Higher Categories

first published

We outline basic concepts of geometric higher category theory and provide links and references for further reading.

## Program overview

Geometric Higher Category Theory is a novel area of research describing combinatorial phenomena arising at the intersection of directed stratified topology and higher algebra. The theory synthesizes notions from classical combinatorial topology with a new combinatorial approach to directedness or ‘framings’. Its aim is to exhibit elements of a unifying language relating, in particular, the following:

• the theory of stratified manifolds (singularity theory, higher Morse theory, classification of smooth structures, etc.),
• quantum algebra (TQFTs, invariants, knots, knotted surfaces and higher knots, etc.),
• and higher category theory (homotopy theory, higher operads, computads, geometric models thereof, etc.).

### Basic ideas and definitions

Many useful ideas are explained in bite-sized chunks in the introductory material listed below.

Readers familiar with the intuition that relates stratified manifolds and higher category theory (in the sense of string diagrams, but generalized to higher dimensions), might want to see definitions straight away, and can do so in this expository paper.

I try to keep this list more or less up-to-date, but I will probably not succeed in doing so; please do leave any remarks about omissions in the comments below.

### Papers, books, theses

Associative $$n$$-categories”, 2018, Christoph Dorn

 “High-level methods for homotopy construction in associative $$n$$-categories”, 2019, David Reutter & Jamie Vicary

Framed combinatorial topology”, 2021, Christoph Dorn & Christopher Douglas

 “Zigzag normalisation for associative $n$-categories”, 2022, Lukas Heidemann, David Reutter & Jamie Vicary

Manifold diagrams and tame tangles”, 2022, Dorn + Douglas

 “Nine short stories about geometric higher categories”, 2023, Dorn

 “From zero to manifold-diagrammatic higher categories”, 2023, Dorn (note: substantial overlap with $n$Lab articles ‘n-truss’ and ‘manifold-diagrammatic n-category’)

 “Globular manifold diagrams”, Dorn + Douglas + Heidemann

In preparation

 “Geometric Computads”, Dorn + Heidemann

In preparation

### Introductory material

Presuming you already know a bit about higher categories, a good comprehensive introduction to geometric higher categories is.

While this essentially summarizes many (but not all) ideas from the list of notes below, the notes are written at a more introductory level (without presuming much about your knowledge of higher categories). Note, the notes are written sequentially (with later notes often referring to earlier ones). The focus is mainly on intuition and brevity, and less on mathematical formality; however, some mathematical substance can nonetheless be found: to indicate this notes are rated with a “formality level” (FL) below.

1. A note on framed computadic and regular cells, explaining the classical problem of defining universal classes of shapes and how to resolve it. (FL 1/5)
2. A note on manifold diagrams, the dual notion of pasting diagrams of framed computadic cells, introduced via a geometric approach. (FL 2/5)
3. A note on two paradigms of higher category theory, explaining syntactically powerful sources of categorical coherences. (FL 1/5)
4. A note on tangles, elementary singularities and higher Morse theory, explaining the close connections of pasting diagrams and the geometry of smooth manifolds. (FL 2/5)
5. A note on the basic combinatorial theory of trusses, leading up to formal defintion of geometric computads, as well as combinatorial defintions of (combinatorial) manifold and pasting diagrams. (FL 4/5)
6. A note on the the formalization of the categorical Pontryagin-Thom construction, relating functors of geometric computads to stratifications on them, and discussing how this should lead to a “combinatorialization” of smooth manifolds and cobordisms. (FL 3/5)