arXiv subjects

Masaki Taho

Publications and source records attributed to Masaki Taho.

5 recordsLinked to original sources

arXiv2026

The Zariski Cotangent Space at the Origin of the Pencil Diffeological Space

The pencil diffeological space is a plane whose smooth structure at the origin is detected only along lines through it. We show that the Zariski cotangent space at the origin is entirely determined by the derivatives along these lines. Moreover, the derivatives along different lines form precisely a $σ$-continuous family, a weak form of continuity. We also compute the external and right tangent spaces at the origin from this description. This suggests that the local smooth structure at a singular point of a diffeological space can give rise to unexpected topological notions.

math.DG↗

arXiv2026

Tangent spaces of diffeological spaces and their variants

Several methods have been proposed to define tangent spaces for diffeological spaces. Among them, the internal tangent functor is obtained as the left Kan extension of the tangent functor for manifolds. However, the right Kan extension of the same functor has not been well-studied. In this paper, we investigate the relationship between this right Kan extension and the external tangent space, another type of tangent space for diffeological spaces. We prove that by slightly modifying the inclusion functor used in the right Kan extension, we obtain a right tangent space functor, which is almost isomorphic to the external tangent space. Furthermore, we show that when a diffeological space satisfies a favorable property called smoothly regular, this right tangent space coincides with the right Kan extension mentioned earlier.

math.AT↗

arXiv2026

Topology and Diffeology via Metric-like Functions

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of metric-like functions originally introduced for spaces of submanifolds. We show that the topologies, uniformities, and diffeologies of these spaces can be systematically derived from the proposed axioms. Furthermore, the framework covers examples such as spaces with compact-open topologies, tiling spaces, and spaces of graphs, which have appeared in different contexts. These results support the study of spaces with metric-like structures from both topological and diffeological perspectives.

math.GN↗

arXiv2026

Diffeological Spaces with a Non-Smooth Derivation

We show that on certain diffeological spaces there exist linear derivations that satisfy the Leibniz rule but are not smooth with respect to the given diffeology. This reveals that the notion of tangent space defined via all such derivations is strictly larger than the one defined using only smooth derivations, showing that smoothness cannot be recovered from the Leibniz rule alone.

math.DG↗

arXiv2025

Infinitely Many Tangent Functors on Diffeological Spaces

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.

math.AT↗