Title: Quotients of Buildings as $W$-Groupoids,
Abstract: We introduce structures which model the quotients of buildings by
type-preserving group actions. These structures, which we call W-groupoids,
generalize Bruhat decompositions, chambers systems of type M, and Tits
amalgams. We define the fundamental group of a W-groupoid, and characterize
buildings as connected simply connected W-groupoids. We give a brief outline of
the covering theory of W-groupoids, which produces buildings as the universal
covers of W-groupoids. The local-to-global theorem of Tits concerning spherical
3-resides allows for the construction of W-groupoids by gluing together
quotients of generalized polygons. In this way, W-groupoids can be used to
construct exotic, hyperbolic, and wild buildings.