Dear all, Yuki Maehara (Tokyo Metropolitan University) will give a talk for this month's PCT seminar (https://pctseminar.github.io/) on Friday September 11 at 10am JST/11am AEST (1am UTC). The zoom link to join the seminar is: https://kyoto-u-edu.zoom.us/j/82401794000?pwd=VjVmzVjmlgxhkdHES4xYeQBTpMzQkS... Title: Towards a homotopy theory of algebraic weak ω-categories Abstract: Mainstream models of weak infinite-dimensional categories (such as quasi-categories or complicial sets) are based on certain non-globular shapes (such as simplices). This allows one to encode the desired category-like *structure* as an existence *property*. For example, composition of 1-cells can be encoded as "for any composable pair f and g, there exists a triangle two of whose edges are f and g", and such a triangle exhibits the third edge as a composite of f and g. We don't require this triangle to be unique for given f and g, so composition is not well defined on the nose. This makes it impossible even to state strict axioms, and hence the resulting model is necessarily weak. But it is not too weak — composition is still "essentially" unique, satisfies the unit and associativity laws "up to equivalence", and so on — thanks to similar existence properties of higher-dimensional shapes. What happens if we instead use the familiar globes so that, for example, the boundary of a 2-cell is given by a parallel pair of 1-cells rather than a triangle? Well, the 2-globe (or in fact the n-globe for any n) doesn't contain a composable pair of 1-cells, so now we must encode composition as an actual algebraic operation rather than as an existence property; this is what "algebraic" means in the title of this talk. I will give an overview of a joint project with Soichiro Fujii and Keisuke Hoshino on such algebraic weak ω-categories, namely those of Batanin and Leinster, which are the Eilenberg–Moore algebras for a suitable monad on the category of globular sets. Compared with the mainstream models (formulated in the language of model categories or ∞-categories), this definition has the downside that it does not naturally come equipped with a homotopy theory, which makes it difficult to capture up-to-equivalence phenomena and constructions. On the other hand, it has the upside that these algebraic weak ω-categories feel much closer to strict ω-categories, particularly because not only the shapes of the cells but also the algebraic structure is designed to mirror the familiar behaviour of their strict counterparts. The overarching theme of this project is therefore to develop a suitable homotopy theory of algebraic weak ω-categories, drawing on the strict case for as much intuition — and even proof strategies — as possible. We hope to see you there! Talks are posted on YouTube afterwards: https://www.youtube.com/@PCTSeminar Best wishes, Soichiro Fujii, Zeinab Galal, JS Lemay