Diffeological vector pseudo-bundles (Q260574)

Diffeology as a subject, introduced by Souriau in the 80's, belongs among various attempts made over the years to extend the usual setting of differential calculus and/or differential geometry. Many of these attempts appeared in the realm of mathematical physics, such as smooth structures à la Sikorski or à la Fröhlicher, and were motivated by the fact that many objects that naturally appear in, for example, noncommutative geometry, such as irrational tori, orbifolds, spaces of connections on principal bundles in Yang-Mills theory, and so on, are not smooth manifolds and cannot be easily treated by similar methods.NEWLINENEWLINEIn the present paper, the author considers a diffeological counterpart of the notion of vector bundle (this counterpart is called a pseudo-bundle). The main difference of the diffeological version is that diffeological pseudo-bundles may not be locally trivial. The paper contains various examples of such, including those where the underlying topological bundle is even trivial. Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), the author considers the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that she indicated, provides when applicable a substitute for said local trivializations. She quickly discusses the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then considers the notion of a pseudo-metric on a diffeological vector pseudo-bundle.