Description
It is generally accepted that the interplay of symmetry and locality in Quantum Field Theory leads one to introduce higher or generalized symmetries. While ordinary (0-form) symmetries form a group, incorporating invertible higher symmetries requires one to replace groups with higher groups, that is, finite connected homotopy types. It is far from obvious how to attach such a gadget to a local QFT. In this talk I discuss this problem in the context of quantum lattice models. I will show how to attach a connected homotopy (d+1)-type to lattice models in d spatial dimensions by exploiting a construction which is a non-abelian analog of the Cech homology of a precosheaf. This homotopy type encodes all higher symmetries as well as all 't Hooft anomalies. A key ingredient in the construction is the equivalence between connected homotopy (d+1)-types and crossed d-cubes of groups due to Loday and Ellis-Steiner.