Homotopical presentations of braid groups via reduced lifts

TL;DR

This paper revisits Deligne's 1997 braid group presentation, extending its validity beyond finite type cases and illustrating its application to actions on Hecke categories.

Contribution

It shows that Deligne's reduced lift presentation applies to all braid groups, not just finite type, and demonstrates its use in constructing braid group actions on Hecke categories.

Findings

Extended Deligne's argument to infinite type braid groups.

Provided a new proof of Dobrinskaya's theorem from 2006.

Illustrated the construction of braid group actions on Hecke categories.

Abstract

In 1997, Deligne showed that the reduced lift presentation of a finite type generalized braid group remains correct if it is (suitably) interpreted as a presentation of a topological monoid. In this expository paper, we point out that Deligne's argument does not require the 'finite type' hypothesis, so it gives a different proof of a theorem proved by Dobrinskaya in 2006. We also review how to use this result to construct an action of the braid group on the finite or affine Hecke $\infty$-category via intertwining functors.