  We describe how an Ore category with a Garside family can be used to
construct a classifying space for its fundamental group(s). The construction
simultaneously generalizes Brady's classifying space for braid groups and the
Stein--Farley complexes used for various relatives of Thompson's groups. It
recovers the fact that Garside groups have finite classifying spaces.
  We describe the categories and Garside structures underlying certain Thompson
groups. The Zappa--Sz\'ep product of categories is introduced and used to
construct new categories and groups from known ones. As an illustration of our
methods we introduce the group Braided T and show that it is of type
$F_\infty$.
