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28
A Tensor Product for GrayCategories
, 1999
"... In this paper I extend Gray's tensor product of 2categories to a new tensor product of Graycategories. 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 2categor ..."
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In this paper I extend Gray's tensor product of 2categories to a new tensor product of Graycategories. 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 2categories. I show that this tensor product of Graycategories satisfies a universal property with respect to quasifunctors of two variables, which are defined in terms of laxnatural transformations between Graycategories. The main result is that this tensor product is part of a monoidal structure on GrayCat, 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 laxnatural transformations, this composite need not be a laxnatural transformation again, making GrayCat only a partial (GrayCat)\Omega  CATegory.
Homotopies and automorphism of crossed modules of groupoids
, 2003
"... Abstract. We give a detailed description of the structure of the actor 2crossed 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 2dimensional holonomy t ..."
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Abstract. We give a detailed description of the structure of the actor 2crossed 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 2dimensional holonomy to be developed elsewhere.
On algebraic models for homotopy 3types
 J. Homotopy Relat. Struct
"... We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. ..."
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We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2.
NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWOFOLD MONOIDAL CATEGORIES
"... 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 EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and a ..."
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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 EckmannHilton 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 twofold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, onearrow 3groupoids (with weak units) cannot realise all simplyconnected homotopy 3types. 1. Introduction and
ON CATEGORICAL CROSSED MODULES
"... Abstract. The wellknown 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 sui ..."
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Abstract. The wellknown 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 2dimensional 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 lowdimensional cohomology categorical groups and, finally, these invariants are connected by a sixterm 2exact sequence obtained by using the kernelcokernel lemma.
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... 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. Th ..."
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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.
NOTES ON 1 AND 2GERBES
, 2006
"... The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. They are for the most part based on the author’s texts [1][4]. Our main goal is to describe the construction which associates to a gerbe or a 2gerbe the corresponding nonabelian coh ..."
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The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. They are for the most part based on the author’s texts [1][4]. Our main goal is to describe the construction which associates to a gerbe or a 2gerbe the corresponding nonabelian cohomology class. We begin by reviewing the wellknown theory for principal bundles and show how to extend this to biprincipal bundles (a.k.a bitorsors). After reviewing the definition of stacks and gerbes, we construct the cohomology class associated to a gerbe. While the construction presented is equivalent to that in [4], it is clarified here by making use of diagram (5.1.9), a definite improvement over the corresponding diagram [4] (2.4.7), and of (5.2.7). After a short discussion regarding the role of gerbes in algebraic topology, we pass from 1 − to 2−gerbes. The construction of the associated cohomology classes follows the same lines as for 1gerbes, but with the additional degree of complication entailed by passing from 1 to 2categories, so that it now involves diagrams reminiscent of those in [5]. Our emphasis will be on explaining how the fairly elaborate equations which define cocycles and coboundaries may be reduced to terms which can be described in the tradititional formalism of nonabelian cohomology. Since the concepts discussed here are very general, we have at times not made explicit the mathematical
On Braidings, Syllepses, and Symmetries
, 1998
"... this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibili ..."
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this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibility. I should mention here that Baez and Neuchl [5, p. 242] (as corrected by me [10, p. 206]) have shown that either one of the functoriality triangles above can be made into an identity, but it is essential to the proof that the other one is not. Defining monoidal 2D teisi as 3D teisi with one object involves a shift of dimension: the arrows, 2arrows and 3arrows of the 3D tas C become the objects, arrows and 2arrows of a 2D tas which will be called the looping of
unknown title
, 2004
"... 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 2gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts [1] and [2]. A notable i ..."
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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 2gerbes. 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 algebrogeometric 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 Gbundle (or right Gtorsor) 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
Journal of Homotopy and Related Structures, vol. 2(2), 2007, pp.119–170 PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES
"... As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to ..."
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As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors 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 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group. 1.