Invertibility and parity in symmetric monoidal categories

TL;DR

This paper introduces a parity concept for morphisms in symmetric monoidal categories, proving a coherence theorem and exploring the structure of super integers with applications to invertible objects.

Contribution

It defines a novel parity notion for morphisms, develops a coherence theorem, and analyzes the super integers within symmetric monoidal categories using 2-monadic algebra.

Findings

Parity is similar to permutation signs but not defined as such.

The work includes a detailed treatment of the free permutative category on an invertible generator.

Examples demonstrate the application of the main results.

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.