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ABSTRACT. 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 £: E -* 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 t(M) over M is associated; t(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 t(M).

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