This is an introductory overview talk about tensor categorical structures behind inclusion of operator algebras. We start with basic tensor categorical concepts behind this theory, and how to understand the standard invariants of subfactors in this language. Based on this we sketch how tensor (or 2-)categories act on operator algebras, extending the case of group actions. These are crucial in understanding the formulation of the celebrated classification theorems for subfactors due to Ocneanu and Popa on the one hand, and also essential for the theory of dimension and probabilistic boundary for unitary rigid tensor categories on the other.
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