We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. The ambient topos makes available some general homotopical machinery.
Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 37-61