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## "Fusion categories and homotopy theory", Pavel Etingof, Dmitri Nikshych, Victor Ostrik, 2010

*Reviewed March 23, 2024*

*Citation:* Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.

*Web:* https://arxiv.org/abs/0909.3140

*Tags:* Mathematical, SPT/SETs, Foundational, Monoidal-categories

In this paper, the authors introduce and prove the pure mathematical
results which underlie the theory of gauging symmetry in topological phases.
One noteworthy feature of this paper, though, is that it makes
no reference to physics or quantum phases. The only stated
motivation for this work is to classify G-graded extensions
of fusion categories, as well as giving a computation-free
derivation of the classification of Tambara-Yamagami fusion categories.

The process of gauging symmetry can be broken into two different steps:

- Turning a group-like symmetry into a categorical symmetry;
- Extending a categorical symmetry into a grading of fusion categories.

In general, both of these steps can fail. That is, there can be obstructions
to the existence of solutions. These obstructions are explained and
are shown to live in certain cohomology classes. In physics
literature, these obstructions are known as *anomalies*.
If there are solutions, then the solutions form a torsor under
a cohomology class living in one dimension lower. Abstractly,
the elements of this torsor are related by *zesting*:

> Delaney, Colleen, et al. "Braided zesting and its applications." Communications in Mathematical Physics 386 (2021): 1-55.

This paper also developed a lot of general fusion category theory,
which was necessary for the application. This theory had to do with
symmetries of fusion categories and modules over fusion categories.