Categories of quantum liquids I

Understand

We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion.

• Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers.
• We also outline a theory of unitary higher categories based on a $*$-version of condensation completion.
• After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases.