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 finding the single-set axioms.

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