The category of piecewise linear spaces. See Rourke and Sanderson. Also: http://www.math.uiuc.edu/K-theory/0012
Julian mentioned something about using PL forms on a simplicial set model of a space, as a replacement for the de Rham cochains, in case the space does not have a smooth structure, in connection with rational homotopy theory.
nLab page on PL category