Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

Comparing the orthogonal and unitary functor calculi

Pages: 227 – 256

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a13


Niall Taggart (Max Planck Institute for Mathematics, Bonn, Germany)


The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.


functor calculus, orthogonal calculus, unitary calculus, G-spectra

2010 Mathematics Subject Classification

55P42, 55P65, 55P91, 55U35

Received 27 April 2020

Received revised 9 November 2020

Accepted 14 December 2020

Published 7 July 2021