We introduce a framework for internal topological symmetries in quantum field
theory, including "noninvertible symmetries" and "categorical symmetries". This
leads to a calculus of topological defects which takes full advantage of
well-developed theorems and techniques in topological field theory. Our
discussion focuses on finite symmetries, and we give indications for a
generalization to other symmetries. We treat quotients and quotient defects
(often called "gauging" and "condensation defects"), finite electromagnetic
duality, and duality defects, among other topics. We include an appendix on
finite homotopy theories, which are often used to encode finite symmetries and
for which computations can be carried out using methods of algebraic topology.
Throughout we emphasize exposition and examples over a detailed technical
treatment.Comment: 65 pages, 37 figures. v2 adds references and corrects topological
misstatements in section 4.4.2. v3 is a major revision with improved
exposition throughout, a new section 2.5 on the passage from local to global
defects, substantially expanded sections 4.4 and A.