Arxiv:1610.07421V4 [Math.AT] 3 Feb 2017 7 Need Also “Commutative Cubes” 12
Modelling and computing homotopy types: I I Ronald Brown School of Computer Science, Bangor University Abstract The aim of this article is to explain a philosophy for applying the 1-dimensional and higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces. Key words: homotopy types, groupoids, higher groupoids, Seifert-van Kampen theorems 2000 MSC: 55P15,55U99,18D05 Contents Introduction 2 1 Statement of the philosophy3 2 Broad and Narrow Algebraic Data4 3 Six Anomalies in Algebraic Topology6 4 The origins of algebraic topology7 5 Enter groupoids 7 6 Higher dimensions? 10 arXiv:1610.07421v4 [math.AT] 3 Feb 2017 7 Need also “Commutative cubes” 12 8 Cubical sets in algebraic topology 15 ITo aqppear in a special issue of Idagationes Math. in honor of L.E.J. Brouwer to appear in 2017. This article, has developed from presentations given at Liverpool, Durham, Kansas SU, Chicago, IHP, Galway, Aveiro, whom I thank for their hospitality. Preprint submitted to Elsevier February 6, 2017 9 Strict homotopy double groupoids? 15 10 Crossed Modules 16 11 Groupoids in higher homotopy theory? 17 12 Filtered spaces 19 13 Crossed complexes and their classifying space 20 14 Why filtered spaces? 22 Bibliography 24 Introduction The usual algebraic topology approach to homotopy types is to aim first to calculate specific invariants and then look for some way of amalgamating these to determine a homotopy type.
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