Appears in collection : Korea-France Joint Workshop on String Theory

In the usual Euclidean path-integral picture, higher symmetries are represented by invertible topological defects, and ’t Hooft anomalies are described by anomaly inflow from a topological action in one dimension higher. Is there a corresponding construction that starts directly from the algebra of local operators and its symmetries? To address this question, I will describe a mathematically precise framework based on localized symmetry transformations and their commutators. To each spatial region U, one associates a group of transformations localized in U. Locality implies that the commutator of transformations localized in regions U and V is localized in their intersection. Organizing these groups and commutator data over covers of space produces a canonical map of homotopy types. Its homotopy fiber is the proposed higher-symmetry type, while composing the map with an ordinary group action gives an obstruction to gauging. The framework is designed to treat algebraic QFT and quantum lattice systems in a common language. The guiding idea is that topology emerges from the interplay of symmetry and locality.