241112 Jaco Ruit

In this talk, I will give a leisurely overview of parts of my Ph.D. research on formal ∞-category theory. Nowadays, Joyal’s and Lurie’s language of ∞-categories has become indispensable in homotopy theory, but it has also found applications in (among others) geometry and representation theory. Beyond the usual ∞-categories, there are now more variants such as enriched and internal ∞-categories. I will explain why it could also be useful to work with these variants. One would like to have access to fundamental categorical concepts for these variants, such as a good notion of (co)limits. I will sketch how one can produce these foundations for all suitable variants of ∞-categories all at once using the theory of so-called ∞-equipments.