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The title of this paper is chosen to imitate that of the paper by van Kampen [10] which gave some basic computational rules for the fundamental group TTX { Y, £) of a based space (an earlier more special result is due to Seifert [14]). In [1] results more general than van Kampen's were obtained in terms of fundamental groupoids. The advantage of the use of groupoids is that one obtains an easy description of the fundamental groupoid of a union of spaces even when the spaces and their intersections are not pathconnected; in such cases, the computation of the fundamental group is greatly simplified by using groupoids. To obtain analogous results in dimension 2 we make essential use of a kind of double groupoid first described in [4]. A major aim is to introduce the homotopy double groupoid p(X, Y,Z) defined for any triple (X, Y,Z) of spaces such that every loop in Z is contractible in Y. The methods of [1] are generalized to give results on p(X, Y,Z). We obtain, as algebraic consequences, results on the second relative homotopy group 7T2{X, Y, t) in the form of computational rules for the crossed module We are grateful to referees for helpful comments.

A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an ‘improvement ’ 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the ‘fundamental 2-group ’ of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern–Simons theory, and are closely related to the Lie 2-algebras g � ( � ∈ R) described in a companion paper. 1 1

We outline the main features of the definitions and applications of crossed complexes and cubical #-groupoids with connections.

In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2-groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kan-fibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...

Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.

We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.

. We present a novel approach to the development of a three-dimensional simplicial refinement strategies based on ideas that have been shown to be effective for two-dimensional refinement. This procedure can be applied with other refinement algorithms with minimal adjustments and can be extended recursively to even higher-dimensional spaces. keywords. mesh refinement, 3D bisection, tetrahedra, adaptivity 1 Introduction As is well known, there are two main steps in local adaptive refinement: the refinement of a subset of elements, and the achievement of the conformity of the mesh. The elements that offer the simplest choice in any dimension are the simplices: triangles in two dimensions, tetrahedra in three dimensions and their analog in even higher dimensions. There are also some algorithms based on bisection or longest-side midpoint insertion [1, 2, 3, 4, 5, 6]. 1.1 Notation and definitions Definition 1 (n-simplex) Suppose X0 ; X1 ; : : : ; Xn are n + 1 points such that the set of ve...

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Dedicated to the memory of Frank Adams