Towards a homotopy theory of algebraic weak $\omega$-categories

At one point during my PhD, my supervisor Dominic Verity told me that he thinks of complicial sets more as simplicial nerves of weak $\omega$-categories than weak $\omega$-categories themselves; if I remember correctly, one of the main reasons he gave was that the simplicial structure is amenable to only one kind of duality.

In that sense, it seems most intuitive to define weak $\omega$-categories as some kind of globular sets. However, one cannot use horn-filling-like properties to encode the desired structure in a globular setting because globes do not have sufficiently many faces. This leads one to algebraic notions of weak $\omega$-category.

In this talk, I will give an overview of our work [2, 3, 4] on algebraic weak $\omega$-categories in the sense of Batanin [1] and Leinster [5], which are the Eilenberg-Moore algebras for a suitable monad (or, more accurately, a suitable globular operad) on the category of globular sets. Compared to more mainstream approaches (formulated in the language of model categories or $\mathrm{\infty }$-categories), one downside of this definition is that it does not naturally come equipped with a homotopy theory. An upside, on the other hand, is that these algebraic weak $\omega$-categories feel much closer to the strict $\omega$-categories, particularly because not only the shapes of the cells but also the algebraic structure is designed to mirror the familiar behaviour of its strict counterpart. Therefore, the overarching theme of this project is to develop a suitable homotopy theory of the algebraic weak $\omega$-categories by drawing as much intuition (and even proof strategies) as possible from the strict case.