Rational orthogonal calculus

Barnes, David

doi:10.1007/s40062-017-0172-4

Cited by 4 publications

(5 citation statements)

References 15 publications

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“…Example 9. 7 The functor nS : J 0 −→ Top * is k-reduced for all k ≤ n. Indeed, the map * −→ S nU is (2n dim(U ) − 1)-connected for all U ∈ J 0 , hence by Lemma 9.5,…”

Section: The Complete Taylor Towermentioning

confidence: 94%

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Unitary calculus: model categories and convergence

Taggart

2022

J. Homotopy Relat. Struct.

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We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

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“…Example 9. 7 The functor nS : J 0 −→ Top * is k-reduced for all k ≤ n. Indeed, the map * −→ S nU is (2n dim(U ) − 1)-connected for all U ∈ J 0 , hence by Lemma 9.5,…”

Section: The Complete Taylor Towermentioning

confidence: 94%

“…The theory of orthogonal calculus has been developed extensively in the literature, for example in [9,31,37] and [7]. Contrastingly, the unitary version does not have solid foundations in the literature despite being known to the experts.…”

Section: Introductionmentioning

confidence: 99%

Unitary calculus: model categories and convergence

Taggart

2022

J. Homotopy Relat. Struct.

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“…With the preliminaries in place, we now construct the S-local orthogonal calculus for S a set of maps of based spaces. Our approach is motivated by the model categorical perspective on orthogonal calculus of Barnes and Oman [6], and Barnes' rational orthogonal calculus [4].…”

Section: Part 2 Local Orthogonal Calculusmentioning

confidence: 99%

“…Proposition 5.1.3 ([6, Proposition 6.5]). There is a model category structure on the category of orthogonal functors with weak equivalences the T n -equivalences 4 and fibrations those levelwise fibrations f : X → Y such the square…”

Section: Polynomial Functorsmentioning

confidence: 99%

The localization of orthogonal calculus with respect to homology

Taggart¹

2021

Preprint

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For a set of maps of based spaces S we construct a version of Weiss' orthogonal calculus which only depends on the S-local homotopy type of the functor involved. We show that S-local homogeneous functors of degree n are equivalent to levelwise S-local spectra with an action of the orthogonal group O(n) via a zigzag of Quillen equivalences between appropriate model categories. Our theory specialises to homological localizations and nullifications at a based space. We give a variety of applications including a reformulation of the Telescope Conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections.

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“…The theory of orthogonal calculus has been developed extensively in the literature, for example in [Wei95, MW09, MW09, BO13] and [Bar17]. Contrastingly the unitary version does not have solid foundations in the literature despite being known to the experts.…”

Section: Introductionmentioning

confidence: 99%

Unitary calculus: model Categories and convergence

Taggart¹

2019

Preprint

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We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra -a variation of orthogonal spectra -as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

show abstract

Rational orthogonal calculus

Cited by 4 publications

References 15 publications

Unitary calculus: model categories and convergence

Unitary calculus: model categories and convergence

The localization of orthogonal calculus with respect to homology

Unitary calculus: model Categories and convergence

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