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Model categories for orthogonal calculus

David Barnes and Peter Oman

Algebraic & Geometric Topology 13 (2013) 959–999

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)–equivariance clearer. Thus we develop model structures for the category of n–polynomial and n–homogeneous functors, along with Quillen pairs relating them. We then classify n–homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)–action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

orthogonal calculus, model categories, spectra, orthogonal spectra
Mathematical Subject Classification 2010
Primary: 55P42, 55P91, 55U35
Received: 1 February 2011
Revised: 10 August 2012
Accepted: 19 September 2012
Published: 5 April 2013
David Barnes
Pure Mathematics Research Centre
David Bates Building
Queen’s University
Belfast BT7 1NN
Peter Oman
Department of Mathematics
Southeast Missouri State University
One University Plaza
Cape Girardeau, MO 63701