Triple Brackets and Lax Morphism Categories |
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Authors: | K A Hardie K H Kamps H J Marcum N Oda |
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Institution: | (1) Department of Mathematics, University of Cape Town, 7700 Rondebosch, South Africa;(2) Fachbereich Mathematik, Fernuniversität, Postfach 940, 58084 Hagen, Germany;(3) The Ohio State University at Newark, 1179 University Drive, Newark, OH 43055, USA;(4) Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka, 814-0180, Japan |
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Abstract: | Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored. |
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Keywords: | Toda bracket (matrix box triple quaternary) 2-category (lax) morphism category homotopy category suspension functor homotopy groups of spheres |
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