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Diffeology, Groupoids & Morita Equivalence

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Diffeology, Groupoids & Morita Equivalence Diffeology, Groupoids & Morita Equivalence M.Sc. Thesis in Mathematics Nesta van der Schaaf 22nd June 2020 Supervisor: Prof. dr. N.P. Landsman Second reader: Dr. I.T. Mărcuț Abstract This thesis consists of two parts. The first (Chapters I and II) is a thorough introduction to the theory of diffeology. We provide a ground-up account of the theory from the viewpoint of plots (in contrast to the sheaf-theoretic treatments). This includes a proof that the category Diffeol of diffeological spaces and smooth maps is complete and cocomplete, (locally) Cartesian closed, and a quasitopos. In addition, we treat many examples, including a detailed recollection of the classification of irrational tori. The second part (Chapters III to VI) is a proposal for a framework of diffeological Morita equivalence. We give definitions of diffeological groupoid actions, -bundles, and -bibundles, general- ising the known theory of Lie groupoids and their corresponding notions. We obtain a bicategory DiffeolBiBund of diffeological groupoids and diffeological bibundles. This has no analogue inthe Lie theory, since we put no ‘principality’ restrictions on these bibundles. We then define a new notion of principality for diffeological (bi)bundles, and subsequently obtain a notion of Morita equivalence by declaring that two diffeological groupoids are equivalent if and only if there existsa biprincipal bibundle between them. Our main new result is the following: two diffeological groupoids are Morita equivalent if and only if they are weakly equivalent in the bicategory DiffeolBiBund. Equivalently, this means that a diffeological bibundle is weakly invertible if and only if it is biprincipal. This significantly generalises the original theorem in the Lie groupoid setting, where an analogous state- ment can only be made if we assume one-sided principality beforehand. As an application of this framework, we prove that two Morita equivalent diffeological groupoids have categorically equival- ent action categories. We also prove that the property of a diffeological groupoid to be fibrating is preserved under Morita equivalence. In a subsequent chapter, we propose an alternative framework for diffeological Morita equivalence using a calculus of fractions. We prove that the notion of Morita equivalence obtained in this way is identical to the one obtained from the bibundle theory. As a corollary, we prove that there is a diffeomorphism between the orbit spaces of two Morita equivalent diffeological groupoids. This generalises the well-known result from the Lie groupoid setting, where in general one only has a homeomorphism. We then give a detailed construction of the germ groupoid of a space, and sketch a theory of atlases. To each atlas on a diffeological space we associate a transition groupoid, capturing the structure of the transition functions between the charts. We prove that two diffeological spaces are diffeomorphic if and only if their transition groupoids are Morita equivalent. This generalises earlier ideas of orbifold atlases to the diffeological setting. ii Preface “What is space?” This was once asked to me in a lecture on the history of quantum mechanics back in 2016. It caught me off guard. Mostly because it had never occurred to me, in any serious sense,to think about it. But it gave me a sense of mysterious excitement1. To me, it was in the same category of questions as “what is a thing?” or “what is truth?” And this thesis, even though not its point to provide an answer, is the closest opportunity I have had to touch upon it. Therefore, I would like to start the Introduction in Chapter I by sketching the landscape of generalised smooth spaces, which should lead naturally into the main topic of this thesis: diffeology. There is something intriguing about the idea that there is an uttermost fundamental structure to it all. Not the form of differential equations or the definition of a smooth atlas, but the very structures that capture those ideas, and then the further structures that underlie those in turn. For me, it is exciting and maybe even comforting to imagine that such a thing exists. During my undergraduate lectures I was fairly close-minded in the way I estimated the broader context of the theories we encountered. The jump from metric spaces to topological spaces was logical, but to think that there is something beyond topology? Something beyond C∗-algebras? Something beyond manifolds? Inside of those theories everything was well-behaved and pleasant, but beyond the boundaries of my ignorance I could not imagine anything else being necessary. Of course, the more I learn, the more these boundaries begin to vanish2. Now, I think a large part of what intrigues me about mathematics is the ways those boundaries can be broken. One of these boundaries has been broken by diffeology, which extends the world of differential topology, reformulating what it means to be smooth. This will be the central notion in this thesis. In Chapter II we provide a detailed introduction to this theory. The main contribution of this thesis describes a generalisation of the theory of Lie groupoids and bibundles to the diffeological setting. In this we get a notion of Morita equivalence for diffeological groupoids. To read more on What this thesis is all about, please refer to Section 1.2. A short note on tangent structures and diffeology. The topic of this project started with groupoids, moved to tangent structures, and then went back to groupoids. Even though the results in this thesis do not relate to tangent structures directly, we nevertheless would like to make some remarks to document some findings that are not part of the main body of this thesis. The motivation for this thesis has its origins in [BFW13] and [Gł19]. There, the authors con- sider embeddings of hypersurfaces in a 4-dimensional Lorentzian manifold that represent solutions to the initial value problem of the Einstein equations. Recalling from [Gł19, Section II.2.1], given a 3-manifold Σ, a Σ-universe is a certain equivalence class of proper embeddings i :Σ ,! (M; g) as space-like hypersurfaces into a Lorentzian 4-manifold. Two embeddings are equivalent when there exists an orientation-preserving isometry on the ambient Lorentzian manifold that sends the image of one embedding to the other. A pair of embeddings (i1; i0), where i1; i0 :Σ ,! (M; g), subject to a similar equivalence relation, then forms a groupoid of Σ-evolutions. The equivalence class of a pair of embed- dings (i1; i0) is interpreted as a Cauchy development of the initial data that Σ represents. [Gł19] proves that the smooth structure on this groupoid defined in [BFW13] makes it into a diffeological groupoid. For physical reasons we want to know the bracket structure of the associated algebroid of Σ- evolutions. These so-called constraint brackets determine the Einstein equations. Even though this ‘algebroid’ is calculated in [BFW13; Gł19], there is no general construction that associates a “diffeolo- gical algebroid” to an arbitrary diffeological groupoid. The first goal of this thesis was then to provide such a construction. It became apparent quite quickly that the foundations for such a construction had yet to be laid. The definition of a Lie algebroid of a Lie groupoid depends heavily on the structure of the tangent bundle 1Not dissimilar to the impressions I got as a teenager when watching documentaries like What the Bleep Do We Know!?, containing quite mystical accounts of quantum mechanics, which in hindsight must have contributed to my choice of going into physics. 2One memorable glimpse of this realisation was during a differential geometry lecture by Gil Cavalcanti. The lecture was about Lie algebras, and the claim was made that the Lie algebra of a diffeomorphism group is the same as the space of vector fields. When a student asked why this was true, we were promptly reassured that the proof wasbeyondthe scope of this course. iii of a smooth manifold, which is not something that we have access to in the diffeological case. In fact, there seems to be no unambiguous notion of tangency on a diffeological space. However, as suggested in [Gł19, Section I.2.3], there is a way to obtain an algebroid-like object for diffeological groupoids, mimicking a technique that is used for Lie groupoids. Namely, [SW15] proves that the Lie algebroid of a Lie groupoid can be obtained as the Lie algebra of the group of bisections. In this sense the question of defining a diffeological algebroid is reduced to defining a diffeological Lie algebra associatedtoa diffeological group3. There has been little work done on this [Les03; Lau11], none of which treats the general case. This leads us to the theory of tangent structures. It was Rosickỳ who first formalised this notion in [Ros84]. A lot of the differential geometric structure of smooth manifolds seemed to be encoded in certain properties of the tangent bundle functor T : Mnfd ! Mnfd, and Rosickỳ was able to condense them into a concise list of functorial and natural conditions. In the two papers [CC14; CC16] the theory is developed in a more modern account, and we refer the interested reader to those papers4. Why is this relevant to our discussion? It appears that the information needed to define a Lie algebra of a Lie group is already encoded in the tangent functor. Generally, the space of vector fields on an object in a tangent category carries a Lie bracket. Therefore, to any group object in such a category we can associate a Lie algebra. Recent work even suggests that we can get algebroids of internal groupoids [Bur17]. Before arriving at a theory of diffeological algebroids, it therefore became clear there was much more work to be done understanding the notion of a tangent structure on diffeological spaces.
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