Cubical $(\omega,p)$-categories

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).