Parity and flexibility

A Picard category is a symmetric monoidal groupoid in which every object has an inverse, up to isomorphism. These structures arise naturally in examples from algebra, geometry, and topology. A natural question to ask is: how does the presence of invertibility change the conclusions of the standard coherence theorem for symmetric monoidal categories? The first answer to this question was given by Dugger in [1] using a rewriting-style method. He proved that it is the parity of the morphisms, rather than their underlying symmetries, that determines if a given diagram commutes.

I have long championed an approach to this coherence theorem through the lens of 2-monad theory. Unfortunately, the “obvious” proof strategy had a gap where two different naturality conditions did not match. In the recent [2], we noticed that this incompatibility can be addressed by using the fact that the free Picard category generated by a single object is flexible as a symmetric monoidal category. In this talk, I will discuss our proof strategy, and if time permits preview ongoing work applying the same methods to braided monoidal structures.