The localization of orthogonal calculus with respect to homology (Q6593003)

The theory of orthogonal calculus, as developed by \textit{M. Weiss} [Trans. Am. Math. Soc. 347, No. 10, 3743--3796 (1995; Zbl 0866.55020)], is concerned with functors from a category of finite-dimensional real inner product spaces to either based topological spaces or orthogonal spectra. The Taylor tower for orthogonal calculus gives suitable approximations to such functors by \(n\)-polynomial functors, and the fibers of the maps in this tower can be considered as \(n\)-homogeneous functors. A model structure for such functors was given by \textit{D. Barnes} and \textit{P. Oman} [Algebr. Geom. Topol. 13, No. 2, 959--999 (2013; Zbl 1268.55001)], and a rational version was established by \textit{D. Barnes} [J. Homotopy Relat. Struct. 12, No. 4, 1009--1032 (2017; Zbl 1386.55012)], reflecting the fact that many applications of orthogonal calculus are done rationally.\N\NIn this paper, the author extends Barnes' result quite generally, providing a version of orthogonal calculus where functors are now considered \(S\)-locally for a designated set \(S\) of maps in the target category. He also produces a model structure for \(n\)-homogeneous functors in this context and provides a Quillen equivalence with \(S\)-local spectra with an \(O(n)\)-action.\N\NA number of applications are given, such as a connection with Bousfield classes, a different approach for the special case of nullifications, an a description of the \(n\)-homogeneous model structure in terms of a homotopy limit of the corresponding model structures for Postnikov sections. Some of the discussion of Bousfield classes can now be further clarified with the disproof of the telescope conjecture. The author also mentions some very interesting potential applications, including to Heuts' extensions of rational homotopy theory to higher chromatic height [\textit{G. Heuts}, Ann. Math. (2) 193, No. 1, 223--301 (2021; Zbl 1481.55007)].