Tangent spaces of bundles and of filtered diffeological spaces

TL;DR

This paper explores the tangent spaces of diffeological bundles and introduces new classes of diffeological spaces, called weakly filtered and filtered, which simplify the understanding of tangent structures and extend existing exact sequences.

Contribution

It introduces the concepts of weakly filtered and filtered diffeological spaces, extending tangent space exact sequences and clarifying tangent bundle structures in these contexts.

Findings

Exact sequence of tangent spaces from a diffeological bundle

Filtered diffeological spaces have more tractable tangent spaces

Hector's tangent bundle coincides with the authors' diffeological tangent bundle in filtered or homogeneous spaces

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.