Two-dimensional homotopical effectivityMPIM

by Jaco Ruit (MPIM)

Thursday Oct 30, 2025, 3:00 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

MPI-Oberseminar

It is a basic observation that equivalence relations on sets correspond bijectively to surjective functions. More generally, one can define equivalence relations in any category, and ask whether this correspondence still holds. An equivalence relation is called effective if it arises as the kernel pair, also known as the Čech nerve, of a morphism. This notion of effectivity was extended to ∞-categories, homotopical analogues of categories, by Lurie, Toën, and Vezzosi, where  equivalence relations are replaced by internal groupoids.

In this talk, I will first elaborate on these ideas and then present ongoing work with Loubaton on a further categorification of effectivity. Specifically, we define a notion of effectivity inside double ∞-categories, a two-dimensional version of ∞-categories. In this setting, internal groupoids are replaced by monoids. I will explain how this double analogue of effectivity can be used to give a conceptual construction of Yoneda embeddings for various generalizations of ∞-categories, such as enriched ∞-categories and ∞-operads, and internal variants.

No prior knowledge of ∞-categories is assumed; we will attempt to work intuitively with this concept throughout the talk.