Invertibility and Parity in Symmetric Monoidal Categories

Abstract

We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers \({\pm }\)1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.