  We consider here the category of diffeological vector pseudo-bundles, and
study a possible extension of classical differential geometric tools on finite
dimensional vector bundles, namely, the group of automorphisms, the frame
bundle, the space of connection 1-forms and the space of covariant derivatives.
Substential distinctions are highlighted in this generalized framework, among
which the non-isomorphism between connection 1-forms and covariant derivatives.
Applications not only include finite dimensional examples with singularities,
but also infinite dimensional vector bundles.
