Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" these two categories is isomorphic to Fr\"olicher spaces, another generalisation structures. then give examples such as well diffeological do not fall into this category. apply theory forms a geometric quotient compact Lie group. subcomplex basic complex quotient. symplectic quotients coming from regular value momentum map, Sjamaar forms. also compare those orbifolds,...

If a Lie group acts on manifold freely and properly, pulling back by the quotient map gives an isomorphism between differential forms basic upstairs. We show that this result remains true for actions are not necessarily free nor proper, as long identity component where space we take in diffeological sense.

We discuss properties of the regular part $S_{reg}$ a subcartesian space $S$. show that is open and dense in $S$ restriction to tangent bundle locally trivial.

We prove that the underlying set of an orbifold equipped with ring smooth real-valued functions completely determines atlas. Consequently, we obtain essentially injective functor from orbifolds to differential spaces.

Differential calculus on Euclidean spaces has many generalisations. In particular, a set $X$, diffeological structure is given by maps from open subsets of to differential $X$ $\mathbb{R}$, and Fr\"{o}licher $\mathbb{R}$ as well $\mathbb{R}$. We illustrate the relations between these structures through examples.

In this paper, we consider Sjamaar’s holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, a number important standard constructions that work for Hamiltonian circle actions in both symplectic category Kähler category: reduction, cutting, blow-up. each case, show theory extends to on complex manifolds with tamed forms. (At least, if fixed points are isolated.) Our main motivation paper is first author needs machinery develop here construct non-Hamiltonian...

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, well "underlying" space any stack. More precisely, so-called concrete sheaves and show that Grothendieck construction sending these to has a left adjoint: functor stack its coarse moduli space. As an application, restrict our attention differentiable examine geometry behind in terms Lie groupoids their principal bundles. Within context, define "gerbe", when groupoid is such gerbe (or represented by...

A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion an submanifold requires additional structure (namely, the choice a topology); when this is unique, we call subset uniquely submanifold. Diffeology provides yet another intrinsic submanifold: diffeological We show that from categorical perspective diffeology rises above others: viewing manifolds as concrete category over sets, initial morphisms are exactly...