Various Notions of Tangent Spaces in Diffeology

Several distinct notions of tangent space for diffeological spaces have been proposed, including the internal and external tangent spaces of Christensen-Wu, as well as alternative constructions due to Iglesias-Zemmour and Vincent. For diffeological spaces, these constructions can differ beyond manifolds. I will compare the main models, use a Kan extension viewpoint to clarify their relationships, and give explicit examples showing that smoothness conditions on derivations can be essential. I will also present a construction of infinitely many pairwise non-isomorphic tangent functors on diffeological spaces. This shows that, beyond manifolds, there is no canonical choice of “the” tangent space, and one must impose additional structural requirements to single out a useful notion.