Categories of quantum liquids I

We develop the mathematical theory of separable and unitary $n$-categories based on Gaiotto and Johnson-Freyd's theory of condensation completion. We use it to study the categories of quantum liquid phases, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-type gapless phases. In particular, we argue that all the topological features of a (potentially gapless) quantum liquid phase can be captured by its topological skeleton. We also show that the category of the topological skeletons of higher dimensional quantum liquid phases can be explicitly computed from a simple coslice 1-category.