Graphical calculus in symmetric monoidal (infty-)categories with duals

Jun Yoshida – 24 May 2017, 7 June 2017

Graphical calculi are a sort of techniques to compute morphisms in monoidal categories, and a really general and geometric formalization was given by Joyal and Street. In this talk, we focus on that in compact categories. Compact categories are examples of pivotal categories, and it is vaguely believed by researchers in quantum representation theory that pivotal categories are described by a calculus of planar tangles. We will give a purely geometric description for this calculus and, using the Cobordism Hypothesis, show every compact category admits a graphical calculus in a coherent way so that we can extend it to the higher contexts.