Single-set cubical categories and their formalisation with a proof assistant (extended version)

We introduce a single-set axiomatisation of cubical \(\omega\)-categories, including connections and inverses. We justify these axioms by establishing a series of equivalences between the category of single-set cubical \(\omega\)-categories, and their variants with connections and inverses, and the corresponding cubical \(\omega\)-categories. We also report on the formalisation of cubical \(\omega\)-categories with the Isabelle/HOL proof assistant, which has been instrumental in developing the single-set axiomatisation.