Elastic diffeological spaces

TL;DR

This paper introduces elastic diffeological spaces, a new class with a natural tangent structure and Cartan calculus, extending smooth manifold concepts to broader contexts like corners, cusps, and mapping spaces.

Contribution

It defines elastic spaces, shows they support a natural tangent functor and Cartan calculus, and includes key examples like manifolds with corners, diffeological groups, and mapping spaces.

Findings

Elastic spaces support a natural tangent functor extending smooth manifold tangent structures.

They form a closed class under coproducts, products, and retracts.

Examples include manifolds with corners, diffeological groups, and mapping spaces.

Abstract

We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural Cartan calculus, consisting of vector fields and differential forms, together with the Lie bracket, de Rham differential, inner derivative, and Lie derivative, satisfying the usual graded commutation relations. Elastic spaces are closed under arbitrary coproducts, finite products, and retracts. Examples include manifolds with corners and cusps, diffeological groups and diffeological vector spaces with a mild extra condition, mapping spaces between smooth manifolds, and spaces of sections of smooth fiber bundles. This paper is a condensed preview of a longer work, explaining its motivation, main concepts, and results, but omitting most of the proofs.