Various Notions of Tangent Spaces in DiffeologyMPIM

by Masaki Taho (University of Tokyo)

Wednesday May 13, 2026, 10:30 AM → 12:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Higher Differential Geometry Seminar

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.