Diffeological differential geometry

The main objective for this thesis is the construction of a tensor bundle on a diffeological space X. Thereby getting access to the exterior bundle of antisymmetric tensors on X, and smooth sections here on i.e. differential forms. We shall list certain requirements that any reasonable tensor bundle on a diffeological space should fulfil. And show that the given construction fulfil these requirements. The main idea of the approach taken in this thesis is to associate to each smooth curve α : R → X a map dα : C∞(X) → R defined by dα(f) := d0(f ◦ α) (where d0 denotes differentiation at 0). This leads to reasonable tangent spaces, tangent bundles, tensor bundles and finally differential forms. These differential forms will in a natural way be D-forms. In order to archive the main objective we shall also need to develop some theory concerning diffeological bundles, and vector bundles.