Handling higher structures such as higher categories usually involves conceiving them as conglomerates of

cells of certain shape. Such shapes include simplices, globes, or cubes. The aim of this talk is to bridge the

gap between two such results:

The first one is the equivalence between cubical and globular $\omega$-groupoids. Although this equivalence

is useful in theory, in practice it is complicated to make explicit the functors composing this equivalence. This

is due to the fact that the proof uses the notion of crossed complexes as a middle ground between globular and

cubical $\omega$-categories.

The second one is the equivalence between cubical and globular $\omega$-Categories.

To this end, we start by studying different notions of invertibility in a cubical $\omega$-category. We then

introduce the notion of cubical $(\omega,p)$-categories, a type of cubical $\omega$-category where all cells of

dimension at least $p+1$ are invertible. In particular, when $p = 0$ or $ p = \omega$, we respectively recover

the notions of cubical $\omega$-groupoids and cubical $\omega$-categories. Lastly, we show that the two

functors forming the equivalence between globular and cubical $\omega$-categories can be restricted to functors

between globular and cubical $(\omega,p)$-categories, and that they still form an equivalence of categories. In

particular, we recover the equivalence between globular and cubical $\omega$-categories in a more explicit fashion.

If time allows, we also show how to extend the following two adjunctions using the notion of $(\omega,p)$-categories:

- The well-known adjunction between globular $\omega$-groupoids and chain complexes
- The adjunction between globular $\omega$-categories and augmented chain complexes (a variant on the notionof chain complexes defined by Steiner).

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