In this paper I extend Gray's tensor product of 2-categories to a new tensor product of Gray-categories. I give a description in terms of generators and relations, one of the relations being an "interchange" relation, and a description similar to Gray's description of his tensor product of 2-categories. I show that this tensor product of Gray-categories satisfies a universal property with respect to quasi-functors of two variables, which are defined in terms of lax-natural transformations between Gray-categories. The main result is that this tensor product is part of a monoidal structure on Gray-Cat, the proof requiring interchange in an essential way. However, this does not give a monoidal (bi)closed structure, precisely because of interchange. And although I define composition of lax-natural transformations, this composite need not be a lax-natural transformation again, making Gray-Cat only a partial (Gray-Cat)\Omega - CATegory.

Abstract. We give a detailed description of the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2-dimensional holonomy to be developed elsewhere.

We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2.

A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of Eckmann-Hilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative two-fold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, one-arrow 3-groupoids (with weak units) cannot realise all simply-connected homotopy 3-types. 1. Introduction and

Abstract. The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernelcokernel lemma for a diagram of categorical crossed modules. We then study derivations with coefficients in categorical crossed modules and show the existence of a categorical crossed module given by inner derivations. This allows us to define the low-dimensional cohomology categorical groups and, finally, these invariants are connected by a six-term 2-exact sequence obtained by using the kernel-cokernel lemma.

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.

A word of warning in lieu of introduction: The aim of the following notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts [1] and [2]. A notable improvement here is that some diagrams in [2] have been refomulated here as (hyper)cubical diagrams, which mirror in the present context certain diagrams introduced by W. Messing and the author in [3]. Since the concepts discussed are very general, it has at times not been made explicit to precisely which mathematical objects they apply. For example, when we refer to “a space ” this might mean a topological space, but also “a scheme ” when one prefers to work in an algebro-geometric context. Similarly, in computing 1 2 cocycles, we will always refer to spaces X endowed with a covering U: = (Ui)i∈I, but the entire discussion remains valid when � i Ui is replaced by a covering morphism Y − → X in an appropriate Grothendieck topology. Finally, there has been no attempt at a serious bibliography, or at making careful attributions of the results mentioned. 1. Torsors and bitorsors Let G be a bundle of groups on a space X. Definition 1.1. A right principal G-bundle (or right G-torsor) on X is a space P π − → X above X, together with a right group action P ×X G − → P of G on P such that the induced morphism

As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group. 1.

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Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures and paracrystals, quantum gravity, as well as the applications of the Goldstone and Noether's theorems to: phase transitions in superconductors/superfluids, ferromagnets, antiferromagnets, mictomagnets, quasi-particle (nucleon) ultra-hot plasmas, nuclear fusion, and the integrability of quantum systems are also considered. Both conceptual developments and novel approaches to Quantum theories are here proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined `machinery ' of