  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.
