2-categorical Opfibrations, Quillen's Theorem B, and $S^{-1}S$

Abstract: Quillen recognized the higher algebraic $K$-groups of a commutative ring $R$ as the homotopy groups of the topological group completion of the classifying space of the category of finitely generated projective $R$-modules. He moreover proved that the topological group completion could be obtained categorically via his $S^{–1}S$ construction. In this talk we will present a 2-categorical version of this result. As part of the proof, we will give a comparison between strict and lax pullbacks for 2-categorical opfibrations, which gives a version of Quillen's Theorem B amenable to applications. This is joint work with Nick Gurski and Niles Johnson.

Mathematics

Audience: researchers in the topic

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