Model categories for orthogonal calculus

Abstract

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.Mathematics Subject Classification: 55P42, 55P91 and 55U35

Rational orthogonal calculus

“…The following result summarises [3,Proposition 6.9]. The weak equivalences of the n-homogeneous model structure may also be described as those maps f such that D n F is an objectwise weak equivalence.…”

“…In this section we recall the construction of the n-polynomial and n-homogeneous model structures from [3]. We start with some basic model category notions.…”

“…The techniques of Bousfield localisation of model categories allow one to construct a new model structure from a given model category with a larger class of weak equivalences. This was used in [3] to construct the n-polynomial and n-homogeneous model structures on J 0 Top from the objectwise model structure. We begin with the necessary terminology.…”

“…For details see [3,Section 6]. In particular that reference shows that the model structure is right proper, despite it being a left Bousfield localisation.…”

“…Working rationally is also central to work of Reis and Weiss [15]. Thus it is natural to ask if one can construct a rationalised version of orthogonal calculus where the tower of a functor F depends only on the (objectwise) rational homology type of F. In this paper we apply the work of [3] to construct suitable model categories that capture the notion of rational orthogonal calculus. In particular, we show that the layers of the rational tower are classified by rational spectra with an action of O(n).…”