Generalizing Quillen's theorem A

Abstract: Quillen's Theorem A, introduced and proved in [2], provides conditions under which a functor $F : C \to D$ of 1-categories induces a weak equivalence $ |N(C)| \to |N(D)|$ of classifying spaces. In this talk, we will discuss two possible ways of generalizing this criterion: working with functors of 2-categories, or finding conditions under which $F$ induces an equivalence between some $\infty$-categorical localizations of $C$ and $D$. Combining these two approaches will lead us to a single generalization of the classical Theorem A to 2-categories equipped with a set of marked morphisms. We will sketch the proof of this generalization provided in [1], and discuss implications.

This work is part of a broader project, aimed at providing computational techniques for $(\infty,2)$-categories in their avatar as scaled simplicial sets. In the next talk, Fernando Abellán García will discuss a related facet of this project, related to (co)limits in $(\infty,2)$-categories.

References:

[1] Fernando Abellán García and Walker H. Stern, Theorem A for marked 2-categories, arXiv:2002.12817.

[2] Daniel Quillen, Higher Algebraic K-theory I: Higher K-Theories, Lecture Notes in Mathematics, vol. 341. Springer, Berlin, Heidelberg.

Mathematics

Audience: researchers in the topic

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