  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.
