A simplicial space M is a separable Hausdorff topological space equipped with an atlas of linearly related charts of varying dimension; for example every polyhedron is a simplicial space in a natural way. Every simplicial space possesses a natural structure complex of sheaves of piecewise smooth differential forms, and the homology of the corresponding de Rham complex of global sections is isomorphic to the real cohomology of M. A cosimplicial bundle is a continuous surjection ξ : E → M \xi :E \to M from a topological space E to a simplicial space M which satisfies certain criteria. There is a category of cosimplicial bundles which contains a subcategory of vector bundles. To every simplicial space M a cosimplicial bundle τ ( M ) \tau (M) over M is associated; τ ( M ) \tau (M) is the cotangent object of M since there is an isomorphism between the module of global piecewise smooth one-forms on M and sections of τ ( M ) \tau (M) .