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

Sorting orders, subword complexes, Bruhat order and total positivity