Abstract
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 Em-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. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all nD quantum liquids form a ∗-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.
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