Invertible cells in $\omega$-categories

TL;DR

This paper investigates the properties of invertible cells in weak ω-categories, providing criteria and algorithms for their invertibility and closure under all operations.

Contribution

It introduces a simple invertibility criterion in computads and an algorithm to compute invertibility data, including inverses and cancellations.

Findings

Invertible cells are closed under all ω-category operations.

A criterion for invertibility in computads is established.

An algorithm computes invertibility witnesses, including inverses and cancellations.

Abstract

We study coinductive invertibility of cells in weak $\omega$-categories. We use the inductive presentation of weak $\omega$-categories via an adjunction with the category of computads, and show that invertible cells are closed under all operations of $\omega$-categories. Moreover, we give a simple criterion for invertibility in computads, together with an algorithm computing the data witnessing the invertibility, including the inverse, and the cancellation data.