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C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables. The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models. It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding. The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).

Physics. --- Quantum Field Theories, String Theory. --- Mathematical Physics. --- Algebra. --- Physique --- Algèbre --- Calculus of tensors. --- Quantum field theory. --- Von Neumann algebras. --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Relativistic quantum field theory --- Algebras, Von Neumann --- Algebras, W --- Neumann algebras --- Rings of operators --- W*-algebras --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Mathematical physics. --- String theory. --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- C*-algebras --- Hilbert space --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis --- Mathematics --- Mathematical analysis --- Physical mathematics --- Models, String --- String theory --- Nuclear reactions

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