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X∗ : X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X. We replace the fundamental group π1(X, ∗) by the homotopy crossed complex πX∗ which consists of the family of groups Cn(p) = πn(Xn, Xn−1, p) for n > 2 and all p ∈ X0, together with the fundamental groupoid C1 = π1(X1, X0) over C0 = X0, all with the standard boundary maps Cn(p) → Cn−1(p) and action of C1. ∗This(More)
The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: mia -> SToft from the category of crossed complexes to the category of spaces. The basic example of a crossed(More)
The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result of [10] was an equivalence of categories γ : G → C. Also established were a version of the homotopy addition lemma, and properties of ‘thin’ elements, in an(More)
Bacterial communities in the airways of cystic fibrosis (CF) patients are, as in other ecological niches, influenced by autogenic and allogenic factors. However, our understanding of microbial colonization in younger versus older CF airways and the association with pulmonary function is rudimentary at best. Using a phylogenetic microarray, we examine the(More)
Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar(More)
A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical(More)
Introduction The fundamental groupoid TT(X) of a topological space X has been known for a long time but has been regarded, usually, as of little import in comparison with the fundamental group—for example, the groupoid is described in ((3) 155) as an 'interesting curiosity'. In this paper we shall generalize the fundamental group at a point a of X, namely(More)