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"An invitation to topological orders and category theory", Liang Kong, Zhi-Hao Zhang, 2022

Reviewed September 13, 2023

Citation: Kong, Liang, and Zhi-Hao Zhang. "An invitation to topological orders and category theory." arXiv preprint arXiv:2205.05565 (2022).

Web: https://arxiv.org/abs/2205.05565

Tags: Expository, Pedagogical, Modular-tensor-categories, Monoidal-categories

This is the first textbook-style account of the use of category theory in topological quantum physics!

This book is definitely written for physicists. A large amount of time is spent discussing the relevant physics, and very little time is spent on the proofs of mathematics results. Surely by the end one would feel comfortable using the language of category theory in their physical work, but one would not feel comfortable enough with the definitions to prove mathematical results.

Additionally, this is by no means a topological quantum computing book. They mention the fact that topological quantum computing exists, but they don't go into any detail about how it is performed.

This is a fantastic reference to be aware of, as well as the bibiliography within. Another good reference, which is perhaps even better for the mathematical inclined audience, is the much earlier PhD thesis

> Bartlett, Bruce H. "Categorical aspects of topological quantum field theories." arXiv preprint math/0512103 (2005).

This PhD thesis seems particularly nice as an introduction to Dijkgraaf-Witten TQFT theory.