What is a homotopy coherent SO(3) action on a 3-groupoid?

One consequence of the cobordism hypothesis is that there's a homotopy coherent action of O(n) on the space of fully dualizable objects in a symmetric monoidal n-category. But what does this mean concretely? For example, by joint work with Douglas and Schommer-Pries there's an O(3) action on the 3-groupoid of fusion categories, but how do you turn this abstract action into a concrete collection of statements about fusion categories? In this talk, I will explain joint work in progress with Douglas and Schommer-Pries answering this question by saying such an action is given by four explicit pieces of data. If time permits I will also discuss homotopy fixed points for SO(3) actions, and explain the relationship between homotopy fixed points and spherical structures on fusion categories.

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