Groupoid Structure Research Articles

Convolution Algebras of Double Groupoids and Strict 2-Groups

Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.

A kind of characterization of homeomorphism and homeomorphic spaces by Core fundamental groupoid: a good invariant

In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it “Core fundamental groupoid” and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., “f : M → N is a homeomorphism if and only if f# 1M → 1N is a topological groupoid isomorphism”. This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid.

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of differentiable groupoids and differentiable stratified groupoids, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for proper locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney (b)-regular stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.

A topological approach to soft groupoids

In this manuscript, the notion of soft topological groupoid is proposed by examining in a different perspective the notion of soft groupoid, which is defined as a parameterized family of the subgroupoids of a soft groupoid. In this context, topological groupoid structures are equipped with soft set theory and the category of soft topological groupoids is formed. Subsequently, the definition of soft topological subgroupoid is presented and all new notions given are strengthened by illustrative examples.

Lifting statistical structures

In this paper, we consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids.

Towards a homotopy domain theory

An appropriate framework is put forward for the construction of \(\lambda \)-models with \(\infty \)-groupoid structure, which we call homotopic \(\lambda \)-models, through the use of an \(\infty \)-category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and \(\lambda \)-calculus, in the sense that the concept of proof (path) of equality of \(\lambda \)-terms is raised to higher proof (homotopy).

REDUCTION OF SYMPLECTIC GROUPOIDS AND QUOTIENTS OF QUASI-POISSON MANIFOLDS

In this work, we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids, which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

We show that the normal form of the Taylor expansion of a $\lambda$-term is isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof along three independent directions. First, we simplify the final step of the proof by following the left reduction strategy directly in the resource calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a groupoid of permutations of copies of arguments in a rigid variant of the resource calculus, and relate the coefficients of Taylor expansion with this structure, while Ehrhard and Regnier worked with groups of permutations of occurrences of variables. Finally, we extend all the results to a nondeterministic setting: by contrast with previous attempts, we show that the uniformity property that was crucial in Ehrhard and Regnier's approach can be preserved in this setting.

Naive cubical type theory

AbstractThis article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman–Hilton duality.

Multiplicative connections and their Lie theory

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn–Drummond [Lie theory of multiplicative tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples.

Homotopies of Crossed Modules of R-Algebroids

In this work, given two crossed modules $$\mathcal {M=}\left( \mu :\mathrm {M} \rightarrow \mathrm {A}\right) $$ and $${\mathcal {N}}=\left( \eta :\mathrm {N} \rightarrow \mathrm {B}\right) $$ of R-algebroids and a crossed module morphism $$f:{\mathcal {M}}\rightarrow {\mathcal {N}}$$ , we introduce an f-derivation as an ordered pair $$H=\left( H_{1},H_{0}\right) $$ of maps $$H_{1}: \mathrm {Mor}\left( \mathrm {A}\right) \rightarrow \mathrm {Mor}\left( \mathrm {N }\right) $$ and $$H_{0}:\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {B} \right) $$ which are subject to satisfy certain axioms and show that f and H determine a crossed module morphism $$g:{\mathcal {M}}\rightarrow \mathcal { N}$$ . Then calling such a pair $$\left( H,f\right) $$ a homotopy from f to g we prove that there exists a groupoid structure of which objects are crossed module morphisms from $${\mathcal {M}}$$ to $${\mathcal {N}} $$ and morphisms are homotopies between crossed module morphisms. Moreover, given two crossed module morphisms $$f,g:{\mathcal {M}}\rightarrow {\mathcal {N}}$$ , we introduce an fg-map as a map $$\varLambda :\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {N}\right) $$ subject to some conditions and then show that $$\varLambda $$ determines for each homotopy $$\left( H,f\right) $$ from f to g a homotopy $$\left( H^{\prime },f\right) $$ from f to g. Furthermore, calling such a pair $$\left( \varLambda ,\left( H,f\right) \right) $$ a 2-fold homotopy from $$\left( H,f\right) $$ to $$\left( H^{\prime },f\right) $$ we prove that the groupoid structure constructed by crossed module morphisms from $${\mathcal {M}}$$ to $${\mathcal {N}}$$ and homotopies between them is upgraded by 2-fold homotopies to a 2-groupoid structure. Besides, in order to see reduced versions of all general constructions mentioned, we examine homotopies of crossed modules of associative R-algebras, as a pre-stage.

Complete Lie algebroid actions and the integrability of Lie algebroids

We give a new proof of the equivalence between the existence of a complete action of a Lie algebroid on a surjective submersion and its integrability. The main tools in our approach are double Lie groupoids and multiplicative foliations, our proof relies on a simple characterization of those vacant double Lie groupoids which induce a Lie groupoid structure on their orbit spaces.

The Lie groupoid analogue of a symplectic Lie group

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the $t$ is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a \textit{symplectic Lie group bundle} (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored.

K-theory and Index Theory for Some Boundary Groupoids

We consider Lie groupoids of the form $${\mathcal {G}}(M,M_1) := M_0 \times M_0 \sqcup H \times M_1 \times M_1 \rightrightarrows M,$$ where $$M_0 = M \setminus M_1$$ and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold $$M_1$$ in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord. They correspond to (the s-connected version of) the problem of the existence of a holonomy groupoid associated to the singular foliation whose leaves are the connnected components of $$M_1$$ and the connected components of $$M_0$$ . We study the Lie groupoid structure of these groupoids, and verify that they are amenable and Fredholm in the sense recently introduced by Carvalho, Nistor and Qiao. We compute explicitly the K-groups of these groupoid’s $$C^*$$ -algebras, we obtain $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}$$ for $$M_1$$ of odd codimension, and $$ K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong \{ 0 \}$$ for $$M_1$$ of even codimension. When M and $$M_1$$ are compact we obtain, as an application of our previous K-theory computations, that in the odd codimensional case every elliptic operator (in the groupoid pseudodifferential calculus) can be perturbed (up to stabilization by an identity operator) with a regularizing operator, such that the perturbed operator is Fredholm; and in the even case, given an elliptic operator there is a topological obstruction to satisfy the previous Fredholm perturbation property given by the Atiyah-Singer topological index of the restriction operator to $$M_1$$ .

Identification of nano-width variants in a fully monoclinic martensitic Ni50Ti50 alloy by scanning electron microscope-based transmission Kikuchi diffraction and improved groupoid structure approach

Nano-width martensite plates in a fully martensitic Ni50Ti50 alloy are indexed successfully by using the off-axis transmission Kikuchi diffraction in scanning electron microscope (i.e., SEM-based TKD). The data obtained by SEM-TKD are effectively interpreted using an improved approach based on the framework of the theoretical groupoid structure method, where the equivalent variants transformed from the monoclinic variants are introduced to calculate all theoretical axis/angle pairs of rotation, and to formulate a complete list of source martensite to target martensite pairs. Consequently, B19′ monoclinic martensite variants in NiTi alloys are identified unambiguously, by using numerical comparison between the experimental and theoretical rotation components, without the reference of retained parent phase.

Symplectic groupoids for cluster manifolds

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type A and X over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered.In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and Mărcuţ. The resulting groupoid structures are also valid in the complex case. These groupoid charts, and their analogues for the symplectic double and blow-up groupoids, are glued by lifting the cluster mutations to groupoid comorphisms whose formulas are motivated by the Hamiltonian perspective of cluster mutations introduced by Fock and Goncharov.

Homotopies of crossed complex morphisms of associative R-algebras

Abstract In this study, given two crossed complexes 𝒞 {\mathcal{C}} and 𝒟 {\mathcal{D}} of associative R-algebras and a crossed complex morphism f : 𝒞 → 𝒟 {f\colon\mathcal{C}\to\mathcal{D}} , we construct a homotopy as a pair ( H , f ) {(H,f\/)} , where H = ( H n ) {H=(H_{n})} is a sequence of R-linear maps H n : C n → D n + 1 {H_{n}\colon C_{n}\to D_{n+1}} . Then we show that for a fixed pair 𝒞 {\mathcal{C}} and 𝒟 {\mathcal{D}} of crossed complexes of associative R-algebras, the family of all homotopies between crossed complex morphisms from 𝒞 {\mathcal{C}} to 𝒟 {\mathcal{D}} has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.

Lie theory of multiplicative tensors

We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorphic structures, obtained through a unifying and conceptual method. We also give a full treatment of multiplicative vector-valued forms, particularly Nijenhuis operators and related structures.

The chromatic symmetry of twins and allotwins.

The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann-Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.

On cocycles

Time dependent physical systems without impulse can be describe by cocycles mathematically. Here we extend the notion of cocycles for Lie groupoids. We consider global pullback absorbing sets and global pullback attractors for them, and we prove that they are invariant under topological equivalence relations which preserve the Lie groupoid structures.