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We study duality of regular weak multiplier Hopf algebras with sufficiently many integrals. This generalizes the well-known duality of algebraic quantum groups. We need to modify the definition of an integral in this case. It is no longer true that an integral is automatically faithful and unique. Therefore we have to work with a faithful set of integrals. We apply the theory to three cases and give some examples. First we have the two weak multiplier Hopf algebras associated with an infinite groupoid (a small category). Related we answer a question posed by Nicolás Andruskiewitsch about double groupoids. Finally, we also discuss the weak multiplier Hopf algebras associated to a separability idempotent. [ABSTRACT FROM AUTHOR]

Previous work (Pradines, C.R. Acad. Sci. Paris 263 (1966) 907, Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2 2 -dimensional information. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to consider the covering morphism and the action of a double groupoid on a groupoid; and then characterize these notions for double group- groupoids. We also give a categorical equivalence between the actions and covering morphisms of double group-groupoids. [ABSTRACT FROM AUTHOR]

We consider the categorical equivalence between crossed modules over groupoids and double groupoids with thin structures, and by this equivalence, we prove how normality and quotient concepts are related in these two categories and give some examples of these objects. [ABSTRACT FROM AUTHOR]

In previous work of the first author and Jibladze, the $$E_3$$ -term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the $$E_3$$ -term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms $$E_r$$ . In this paper, we introduce 2-track algebras and tertiary chain complexes, and we show that the $$E_4$$ -term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations. [ABSTRACT FROM AUTHOR]

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, C R Acad Sci Paris 256:1198-1201, , Ann Sci Ec Norm Super 80:349-425, ) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural 'filling condition'. Any such double groupoid characteristically has associated to it 'homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of 'weak equivalence' between such double groupoids, and a corresponding 'homotopy category' is defined. Our main result in the paper states that the geometric realization functor induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy 2-types (that is, the homotopy category of all topological spaces with the property that the $n{\text{th}}$ homotopy group at any base point vanishes for n ≥ 3). A quasi-inverse functor is explicitly given by means of a new 'homotopy double groupoid' construction for topological spaces. [ABSTRACT FROM AUTHOR]

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid. [ABSTRACT FROM AUTHOR]

Abstract: In the context of non-abelian gerbes, we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We address their two-dimensional parallel transport, study its properties, and construct non-abelian Wilson surface functionals. [ABSTRACT FROM AUTHOR]

Abstract: The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and quantum and Frobenius–Perron dimensions of simple objects are computed. [Copyright &y& Elsevier]

Previous work (Pradines, C. R. Acad. Sci. Paris 263 (1966) 907; Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information. [Copyright &y& Elsevier]