Quantum topology without topology
Lecture notes explaining quantum topology from the categorical-algebra side: diagrams first, topological meaning emerging along the way.
Data
- Title: Quantum topology without topology
- Authors: Daniel Tubbenhauer
- Status: preprint. Last update: Fri, 13 Jun 2025 07:56:18 UTC
- Code and (possibly empty) Erratum: Click
- ArXiv link: https://arxiv.org/abs/2506.18918
Abstract
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum physics helping to transfer ideas, and stimulating mutual development. They also possess deep and intriguing connections to representation theory, particularly through representations of quantum groups. These lecture notes aim to illustrate how categorical algebra provides a framework for studying both algebra and topology. Specifically, they demonstrate how quantum invariants emerge naturally from a mostly categorical perspective.
What is the point?
These notes are meant as a guided route into quantum topology through categorical algebra. Instead of starting with all the topology, they build the diagrammatic and categorical machinery first, and then explain how quantum invariants arise from it.
Categories, monoidal categories, pivotal and braided structures.
Quantum invariants from diagrammatic and web-based viewpoints.
A route between topology, algebra, number theory and quantum physics.
A picture
A visual map of the categorical Rosetta stone used throughout the notes.
A few extra words
The 13 lectures (and one bonus lecture) are:
- Categories – definitions, examples and graphical calculus
- Monoidal categories I – definitions, examples and graphical calculus
- Monoidal categories II – more graphical calculus
- Pivotal categories – definitions, examples and graphical calculus
- Braided categories – definitions, examples and graphical calculus
- Additive, linear and abelian categories – definitions and examples
- Fiat and tensor categories – enrich the concepts from before
- Fiat, tensor and fusion categories – definitions and classifications
- Fusion and modular categories – definitions and graphical calculus
- Quantum invariants – a diagrammatic approach
- Quantum invariants – a web approach
- Growth in monoidal categories
- How good are quantum invariants? – a big data approach
- Appendix A. Representation theory of finite dimensional algebras with MAGMA