Re-innovating the wheel: Tannaka duality and extended operators in QFT

Tannaka duality was developed in the late 1930’s in order to reconstruct a complex Lie group from its category of representations. Its generalisation, more often simply called “reconstruction theory,” has now become a fundamental part of representation theory — in particular, representation theory of categories and higher categories. It was used in physics in the 1990’s (and the 2000’s… and again in the 2010’s…) to represent line operators in topological QFT’s — by folks like Freed, Witten, and Costello. I will talk about how I stumbled into it in the course of my work, was drawn in by its power, and have been seeking new applications.