Pedro Resende
This is a short survey paper, partly meant as a research announcement. Its purpose is to highlight some aspects of the interplay between quantales, inverse semigroups, and groupoids. Many of the results mentioned have not yet been presented and will appear in detail elsewhere.
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We provide three functorial extensions of the equivalence between localic etale groupoids and their quantales. The main result is a biequivalence between the bicategory of localic etale groupoids, with bi-actions as 1-cells, and a bicategory of inverse quantal frames whose 1-cells are bimodules. As a consequence, the category InvQuF of inverse quantale frames, whose morphisms are the (necessarily involutive) homomorphisms of unital quantales, is equivalent to a category of localic etale groupoids whose arrows are the algebraic morphisms in the sense of Buneci and Stachura. We also show that th…
We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic étale groupoid $G$) together with a map of involutive quantales $p:Q\to\mathcal L^{\bigvee}(S)\ [=\mathcal O(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of étale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.
For an arbitrary localic etale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of etendues.
Taking advantage of the quantale-theoretic description of étale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras.
We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of etale groupoid is subsumed in a natural way by that of quantale. In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic etale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distr…
Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and étale groupoids. Here we study a further generalization where a non-unital version of supported quantale carries the algebraic content of such correspondences to the setting of open groupoids. A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.
It is well known that inverse semigroups are closely related to étale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic étale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially…
Taking into account the correspondence between open groupoids and their quantales, we establish a bijective correspondence between the principal $G$-bundles whose left projection is an open surjection and the principal $\opens(G)$-locales for any open groupoid $G$ and its groupoid quantale $\opens(G)$, the latter requires a generalization of the supported modules for groupoid quantales.
By restricting to a class of localic open groupoids $G$ which, similarly to Lie groupoids, possess appropriate covers $\widehat G\to G$ by étale groupoids, we extend results about groupoid actions and quantales that were previously proved for étale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of $G$-actions as a category of quantale modules on $\mathcal O(\widehat G)$ that satisfy a condition related to the quantale $\mathcal O(G)$. This leads to a simple description of $G$-sheaves and the classifying topos of $G$ in…
Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic eta…
In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375], we have discussed the correspondence between two notions of a gerbe over a stack. In [arXiv:2012.08447], we have developed the notion of Chern-Weil map for principal bundles over a Lie groupoid when the Lie groupoid is equipped with a connection. In [arXiv:2012.08442], we discuss the notion of connection on the principal bundle over a Deligne Mumford stack …
The representing localic groupoid for a geometric theory
本文通俗阐述了每个 topos 可表示为 locale 群胚上层 topos 的 Joyal-Tierney 结果，并明确给出了给定几何理论分类 topos 的表示 locale 群胚展示及其与 Forssell 拓扑群胚的联系。
Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models…
The holonomy Lie $\infty$-groupoid of a singular foliation I
本文提出通过双子浸入整合奇异叶形的有限维更高李群体，并将其1-截断为Androulidakis-Skandalis holonomy群体。
This is a survey concerning the relationship between Lie Groupoids (and their morphisms) and singular foliations in the sense of Sussmann-Stefan (considered from a purely geometrical point of view). We focus on the interaction between the algebraic and differentiable structures underlying Lie groupoids, and between groups and graphs of equivalence relations, regarded as two basic degeneracies for groupoids. Historical remarks, motivations and examples are developed in five appendices.
Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrcun, Cattaneo-Felder and Crainic-Fernandes, among others. In this thesis we study - part of - the categorified version of this story, namely the integrability of LA-groupoids (groupoid objects in the category of Lie algebroids), to double Lie groupoids (groupoid objects in the category of Lie groupoids) providing a first set of sufficient condi…
Generic bundles over a localic category
本文构造了分类局部范畴和群胚，用于带有逻辑结构的各种丛，并证明了局部群胚比拓扑斯能分类更多种类的逻辑理论。
A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects
本文通过几何方法构建了带有所有维数缺陷的3维未扭Dijkgraaf-Witten理论，利用对称单子函子将缺陷cobordism范畴映射到复向量空间范畴，无需三角化状态和更高范畴的图示计算。
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