Marked colimits and higher cofinality

Abstract: In this talk I will introduce the notion of marked colimits in $\infty$-bicategories, providing a natural framework to interpret Theorem A$^\dagger$ from the previous talk as a higher cofinality statement. Relevant examples will be discussed, as a well the relation of marked colimits with weighted colimits. The main result that I will discuss is the following:

Let $C^\dagger$, $D^\dagger$ be $\infty$-categories equipped with a collection of marked morphisms and let $f$ be a functor from $C^\dagger$ to $D^\dagger$ that preserves the marking. Then $f$ is a marked cofinal functor (i.e., restriction along $f$ preserves marked colimits) if and only if the conditions of Theorem A$^\dagger$ are satisfied.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)