Diffeological connections on diffeological vector pseudo-bundles

TL;DR

This paper introduces a new, simpler definition of diffeological connections on pseudo-bundles, adapting standard concepts to the diffeological setting using dual pseudo-bundles for tangent vectors.

Contribution

It proposes a straightforward, alternative definition of diffeological connections based on smooth sections and dual pseudo-bundles, differing from previous approaches.

Findings

The new definition simplifies the concept of diffeological connections.

It explores the interaction of smooth sections with gluing constructions.

The space of smooth sections may be infinite-dimensional, even for trivial bundles.

Abstract

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections. One aspect prominent in the diffeological context has to do with the choice of an appropriate substitute for tangent vectors and smooth vector fields, since there are not yet standard counterparts for these notions. In this respect we opt for the simplest possibility; since there is an established notion of the (pseudo-)bundle of differential forms on a diffeological space, we take the corresponding dual pseudo-bundle to play the role of the tangent bundle. Smooth vector fields are then smooth sections of this dual pseudo-bundle; this is one reason why we devote a particular attention to the space of smooth sections of an arbitrary diffeological vector pseudo-bundle (one curiosity is that it might easily turn out to be infinite-dimensional, even when the pseudo-bundle itself has a trivial finite-dimensional vector bundle as the underlying map). We concentrate a lot on how this space interacts with the gluing construction for diffeological vector pseudo-bundles (described in [10]). We then deal with the same question for the proposed notion of a diffeological connection.