Infinitely Many Tangent Functors on Diffeological Spaces
Masaki Taho
TL;DR
This work shows that tangent functors on diffeological spaces are far from unique once one leaves the realm of smooth manifolds. By introducing two families of functors parameterized by test spaces, $T^{(Y,y)}$ and $\hat{T}^{(Y,y)}$, and analyzing concrete test spaces such as irrational tori $T_\alpha$ and orbit spaces $H_n$, the authors prove the existence of uncountably many pairwise non-isomorphic internal tangent functors and infinitely many right-type tangents. They establish clear nontriviality criteria (e.g., when a map between test spaces exists, a tangent component is nonzero) and show that even within known constructions (internal, right, Vincent-type) there is a rich landscape of possible tangent-like structures, depending on the chosen test space. The results highlight a fundamental sensitivity of tangent functors to the ambient diffeological framework, with potential implications for differential-geometric constructions beyond smooth manifolds.
Abstract
We study tangent spaces in the setting of diffeological spaces. Several distinct tangent functors have been introduced, each of which extends the classical tangent functor from smooth manifolds. In this paper, we construct infinitely many non-isomorphic tangent functors on diffeological spaces. We compare our constructions with existing models, including the internal and external tangent spaces. Our results show that the choice of tangent functor is far from unique outside smooth manifolds.