Differential geometry on simplicial spaces

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 $\xi :E \to M$ 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 $\tau (M)$ over M is associated; $\tau (M)$ 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 $\tau (M)$.