  In this note we construct a poset map from a Boolean algebra to the Bruhat
order which unveils an interesting connection between subword complexes,
sorting orders, and certain totally nonnegative spaces. This relationship gives
a new proof of Bj\"orner and Wachs' result \cite{BW} that the proper part of
Bruhat order is homotopy equivalent to the proper part of a Boolean algebra ---
that is, to a sphere. We also obtain a geometric interpretation for sorting
orders. We conclude with two new results: that the intersection of all sorting
orders is the weak order, and the union of sorting orders is the Bruhat order.

Combinatorics;Algebraic Geometry