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A recent paper [R22] established "Frobenius reciprocity" as a bijection $t$ between certain symplectically reduced spaces (which need not be manifolds), and conjectured: 1{\deg}) $t$ is a diffeomorphism when these spaces are endowed with their natural subquotient diffeologies, 2{\deg}) $t$ respects the reduced diffeological $2$-forms they may (or might not) carry. In this paper, we prove both this conjecture and a similar one on prequantum reduction, and also give new sufficient conditions for the reduced forms to exist. We stop short of proving that they always exist., Comment: 26 pages

There are several notions of a smooth map from a convex set to a cartesian space. Some of these notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each $k$ extends to a $C^k$ function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces., Comment: 16 pages, 1 figure; version 2 has minor changes; to appear in AMS Contemporary Mathematics

We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences \`a la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors \`a la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences., Comment: 27 pages; v2 is a version more appropriate for publication, and is streamlined. However, v1 has more details for many of the proofs

We study three variants of bicategories of diffeological groupoids, and show that they are all equivalent: the bicategory with generalised morphisms, the bicategory with anafunctors, and the bicategory with bibundles. Moreover, we show that the corresponding sub-bicategories of locally subductive diffeological groupoids, as well as Lie groupoids, are (essentially) full sub-bicategories. As applications, we obtain equivalences with the corresponding 2-categories of geometric stacks, and present some Morita invariants., Comment: 84 pages; v2 is essentially the same as v1 with some small modifications to the introduction and typographical corrections

A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a uniquely immersed submanifold. Diffeology provides yet another intrinsic notion of submanifold: a diffeological submanifold. We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the initial morphisms are exactly the (diffeological) inductions, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of pseudo-immersions. We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective. In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold., Comment: 15 pages, 3 figures

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their \v{C}ech cohomology. As an application, we prove that the first degree \v{C}ech cohomology classes for the sheaf of smooth functions to an abelian diffeological group $G$ classify the diffeological principal $G$-bundles., Comment: 28 pages; v2: besides minor corrections and additions, we also added a subsection comparing this cohomology with that of Iglesias-Zemmour

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces., Comment: 15 pages; to appear in Proceedings of the AMS; V3 has modified structure and added applications

Differential calculus on Euclidean spaces has many generalisations. In particular, on a set $X$, a diffeological structure is given by maps from open subsets of Euclidean spaces to $X$, a differential structure is given by maps from $X$ to $\mathbb{R}$, and a Fr\"{o}licher structure is given by maps from $\mathbb{R}$ to $X$ as well as maps from $X$ to $\mathbb{R}$. We illustrate the relations between these structures through examples., Comment: 37 pages; v2 has more examples, results on underlying topologies, and an appendix summarizing relationships with other smooth structures

This paper exhibits equivalences of 2-stacks between certain models of $\mathbb{S}^1$-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to $\mathbb{S}^1$-bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting of $\mathbb{S}^1$-gerbes over Lie groupoids., Comment: 23 pages

In this paper, it is shown that non-isomorphic effective linear circle actions yield non-diffeomorphic differential structures on the corresponding orbit spaces., Comment: 17 pages. This paper is the result of a Summer 2016 REU (undergraduate research) project at the University of Colorado Boulder. [V2 update: the earlier version contained material not dealt with during the REU, which will be published separately.] To appear in Involve

Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $\Psi$. Fix a value $a$ of $\Psi$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group action. We prove under mild assumptions that this only occurs if the symplectic quotient is diffeomorphic to an effective orbifold. This, in turn, only occurs if $a$ is a regular value, or there is at most one positive weight or at most one negative weight., Comment: 20 pages V2: added an assumption "weakly unrepresentability" to account for a gap in V1

We define a diffeology on the Milnor classifying space of a diffeological group $G$, constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori., Comment: 29 pages -- v3 fixes another error, and clarifies some comments on differential forms

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

We prove that the underlying set of an orbifold equipped with the ring of smooth real-valued functions completely determines the orbifold atlas. Consequently, we obtain an essentially injective functor from orbifolds to differential spaces., Comment: V3: Final version. Removed a superfluous "2" from some examples

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

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Within this context, we define a "gerbe", and show when a Lie groupoid is such a gerbe (or when a stack is represented by one). Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space., Comment: 19 pages; v5: added an application on gerbes (to appear in AMS Contemporary Mathematics). v4: besides some minor changes, the abstract was rewritten and we added an application to the differentiable stack section. v3: condensed the presentation and added more information to the introduction. (Also changed the title.)

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space., Comment: To appear in Trends in Mathematics, Research Perspectives: Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019, Birkh"auser. V4: Many straightforward details removed, shortening the paper to 9 pages. Two new corollaries of the main theorem given. Terminology updated, using the language of del Hoyo-Fernandes

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action., Comment: V1 is the official PhD thesis - University of Toronto. V2: Some minor changes, including some acknowledgements for two important lemmas and a corollary. 108 pages

Let $G$ be a Lie group, and let $(M,\omega)$ be a symplectic manifold. If $G$ admits a Hamiltonian action on $(M,\omega)$ with momentum map $\mu$, then $M$, the zero-level set of $\mu$, the orbit space, and the corresponding symplectic quotient all have induced stratifications. We push this setting into the language of differential spaces, and as a consequence we find that the stratifications are intrinsic to the ring of smooth functions on each space., Comment: 44 pages. Earlier versions of this paper were supposed to prove a result regarding a de Rham complex of differential forms on the symplectic quotient. A crucial lemma in the proof was incorrect. The theory used remains useful for studying the stratifications mentioned in the abstract from the point of view of differential spaces. Erroneous parts removed

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth., Comment: 16 pages

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