  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.
