Link to talk: http://andreghenriques.com/VariousThingsActedOnByFusionCategories.mp4
Besides von Neumann algebras and C*-algebras, there exist a couple of other mathematical object for which can be acted upon by fusion categories. These include: spin chains; conformal nets; and, conjecturally, cocycles for elliptic cohomology. I will review, in all of these cases, what it means for a fusion category to act, and comment on the implications.
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