Groupoid Structure Research Articles

From simplicial homotopy to crossed module homotopy in modified categories of interest

Abstract We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

A gentle introduction to Schwinger’s formulation of quantum mechanics: The groupoid picture

In this paper, we review Schwinger’s formulation of Quantum Mechanics and argue that the mathematical structure behind Schwinger’s “Symbolism of Atomic Measurements” is that of a groupoid. In this framework, both the Hilbert space (Schrödinger picture) and the [Formula: see text]-algebra (Heisenberg picture) of the system turn out to be derived concepts, that is, they arise from the underlying groupoid structure.

Higher cohomologies of commutative monoids

Extending the Eilenberg–Mac Lane approach, we introduce and explore higher-level cohomology theories for commutative monoids and compare them with pre-existing theories (Leech, Grillet, etc.). We offer a cohomological classification of symmetric monoidal groupoid structures and work out some explicit computations for cyclic monoids.

On groupoid gradings

Abstract We introduce the notion of groupoid grading, give some nontrivial examples and prove that groupoid gradings on simple commutative or anti-commutative algebras are necessarily group gradings. We also take advantage of the structure of groupoids to prove some results about groupoid gradings and certain coarsenings of these which turn out to be group gradings. We also study set gradings on arbitrary algebras, by characterizing their homogeneous semisimplicity and their homogeneous simplicity in terms of a property satisfied by the supports of the gradings, and also relate set gradings with groupoid gradings via coarsenings. Finally we study a class of set gradings on M n ( C ) , the orthogonal gradings, and show that all of them which are fine are necessarily groupoid gradings.

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

On the identity type as the type of computational paths

We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is easy to understand and use. We make this point clear constructing terms of some relevant types using our proposed elimination rule. We also show that the identity type, as dened by our approach, induces a groupoid structure. This result is on par with the fact that the traditional identity type induces a groupoid, as exposed by Hofmann & Streicher (1994).

Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes.

The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral. (2016), 28, 189-203], has been performed on the respective labelled quotient graphs (LQGs). It is shown that the structure and maximum symmetry of the module, i.e. its layer group, can be determined directly from the LQG. Partial symmetry operations between different modules have been associated with automorphisms of the quotient graph that may not be consistent with net voltages over the respective cycles. These operations have been shown to generate the pyroxene groupoid structure.

Wild ramification and the cotangent bundle

We define the characteristic cycle of a locally constant étale sheaf on a smooth variety in positive characteristic ramified along the boundary as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary. The crucial ingredient in the definition is the commutative group structure on the boundary induced by the groupoid structure of multiple self-products. We prove a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the cohomological characteristic class under a certain condition and a concrete example where the intersection with the 0 0 -section computes the Euler-Poincaré characteristic.

The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces

AbstractA surface ∑ endowed with a Poisson tensor π is known to admit a canonical integration, 𝒢(π), which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if π is not an area form on the 2-sphere, then 𝒢(π) is diffeomorphic to the cotangent bundle T*∑. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

Multiplicative integrable models from Poisson–Nijenhuis structures

We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].

Quantization of Poisson Manifolds from the Integrability of the Modular Function

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, combining the tools of geometric quantization with the results of Renault’s theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular, we consider the case when the modular function is multiplicatively integrable, i.e., when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on $${\mathbb{C}P_n}$$ , seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU(n + 1). We show that a bihamiltonian system on $${\mathbb{C}P_n}$$ defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu’s description of quantum homogeneous spaces as groupoid C*-algebras.

The transformation groupoid structure of the q-Gaussian family

Abstract Groupoid theory plays an important role in physics since the beginnings of quantum mechanics. Recent developments in understanding symmetries in complex dynamical systems underpin the growing importance of groupoid theory also for statistical mechanics. The q -Gaussian function is observed as the distribution function of many physical and biological systems and emerges naturally in the statistical mechanics of non-ergodic and complex systems. A number of dynamical systems are characterized by pairs and triples of q -Gaussians. The aim of this work is to relate these triples of q -Gaussians with different q -values, representing intrinsic symmetries of the dynamical system at hand, such that any value of q can be mapped uniquely to any other value q ′ . We present a complete set of transformations of q -Gaussians by deriving a general map γ q q ′ that transforms normalizable q -Gaussian distributions into one another. We show that the action of γ q q ′ is a transformation groupoid.

Deformations of affine varieties and the Deligne crossed groupoid

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R=K〚ℏ〛). We consider associative (resp. Poisson) R-deformations of the structure sheaf OX. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of OX. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex.The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.

Structure and classification of monoidal groupoids

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech’s cohomology groups of monoids.

Moduli spaces of flat connections and Morita equivalence of quantum tori

We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3-dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.

The leaf space of a multiplicative foliation

We show that if a smooth multiplicative subbundle$S\subseteq TG$ on a groupoid $G⇉P$ is involutive and satisfies completeness conditions, then its leaf space$G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown toproject by forward Dirac maps to Poisson groupoids.

GROUPOID ALGEBRAS ASSOCIATED WITH COVERING MAPS

For a compact Hausdorff space X with its p-fold covering map <TEX>${\sigma}$</TEX>, we construct its corresponding topological groupoid <TEX>${\Gamma}$</TEX>, and show that there is a strong relation between the dynamical structures of (X, <TEX>${\sigma}$</TEX>) and the groupoid structures of <TEX>${\Gamma}$</TEX>.

Some remarks on the global structure of proper Lie groupoids in low codimensions

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G ⋉ X with G compact.

Formal Lagrangian Operad

Given a symplectic manifold M, we may define an operad structure on the the spaces Ok of the Lagrangian submanifolds of via symplectic reduction. If M is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semiclassical part of Kontsevich′s deformation of C∞(ℝd) is a deformation of the trivial symplectic groupoid structure of T∗ℝd.

Automorphisms and Homotopies of Groupoids and Crossed Modules

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids \(\mathcal{C}\), in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups \(\mathcal{C}_u\) formed by restricting to a single object u. Finally, we show that the group of homotopies of \(\mathcal{C}\) may be determined once the group of regular derivations of \(\mathcal{C}_u\) is known.