Tangent spaces of bundles and of filtered diffeological spaces

Abstract

Abstract. We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle T H X defined by Hector is a diffeological vector space over X when X is filtered or when X is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper.

Diffeological vector pseudo-bundles

“…Note that, with the way this definition is stated, we need to explain why it makes sense; more precisely, why the pseudo-bundle diffeology exists. It does for essentially the same reason (having to do with the lattice property of diffeologies, see [6], Section 1.25), which is already explained in [1] (see Proposition 4.6, cited also in the present paper, Sect. 3.1).…”

“…In the rest of the paper we opt for the term diffeological vector pseudo-bundle to denote the same object that is called a regular vector bundle in [13] and a diffeological vector space over X in [1] (the definition of which we cited in the previous section). We avoid the former term to distinguish our objects of interest from true diffeological vector bundles (that are locally trivial), while the term of ChristensenWu can be confused with a diffeological vector space proper (that is, a vector space endowed with a vector space diffeology); besides, it requires to introduce a notation for the base space, something which on occasion might be superfluous or cumbersome.…”

“…16 It is not immediately clear whether this pullback would necessarily make V into a diffeological vector space over X; however, whichever the case, there is a standard way, due to Christensen-Wu [1], to make it into such, described later in the paper.…”

“…A trivial extension of this concept just requires substituting each mentioning of a smooth map with "diffeologically smooth", and maybe choosing a diffeology on the fibre (but then, in the finite-dimensional case, which we are limiting ourselves to, there is a standard choice). Yet, it is easily seen that this is not sufficient; one way to to explain why is to point to the internal tangent bundles of Christensen-Wu [1]. These, frequently enough, turn out not to be vector bundles at all, for the simple reason that they do not always have the same fibre (for reasons related to the underlying topology of the base space), yet, they are more than legitimate candidates to be tangent bundles.…”

“…Thus, the aim of this note is to take a closer look at the objects of this type, starting from attempting to define with precision what "of this type" means (apart from the very general definition given in [1]). In particular, we explore the path of constructing these "pseudo-bundles" (as we call them) by a kind of…”