Tangent spaces of diffeological spaces and their variants

TL;DR

This paper explores different methods of defining tangent spaces in diffeological spaces, focusing on the relationship between the right Kan extension and external tangent spaces, and introduces a modified right tangent space functor.

Contribution

It introduces a modified right tangent space functor for diffeological spaces and clarifies its relationship with the external tangent space and the right Kan extension.

Findings

Modified right tangent space is almost isomorphic to the external tangent space.

When the space is smoothly regular, the right tangent space coincides with the right Kan extension.

Provides new insights into tangent space constructions in diffeological spaces.

Abstract

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.