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).
