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Symplectic Weiss calculi

Matthew Carr, Niall Taggart

TL;DR

The paper develops two symplectic Weiss calculi: a non-compact version based on symplectic vector spaces, which deformation retracts onto unitary calculus via the groupoid-core equivalence, and a compact quaternion version built with quaternionic Stiefel combinatorics that parallels and contrasts unitary and orthogonal calculi. The deformation retraction is established by comparing indexing categories through Quillen's bracket construction and showing that the unitary and symplectic calculi are equivalent at the level of ∞-categories and their Weiss towers. In the quaternionic setting, derivatives are classified by spectra with Sp(n) action, yielding a precise Homog^n(J^H,S_*) ≃ Sp^{B Sp(n)} correspondence and a robust tower formalism. Applications include the classification of stably trivial quaternionic vector bundles over finite complexes, described via stable maps to threefold suspensions of stunted quaternionic projective spaces, and a detailed comparison framework linking quaternionic calculus with unitary and orthogonal calculi. The findings illuminate the universality of Weiss calculus across symplectic geometries and provide concrete computational tools for stable quaternionic bundles and their moduli.

Abstract

We provide two candidates for symplectic Weiss calculus based on two different, but closely related, collections of groups. In the case of the non-compact symplectic groups, i.e., automorphism groups of vector spaces with symplectic forms, we show that the calculus deformation retracts onto unitary calculus as a corollary of the fact that Weiss calculus only depends on the homotopy type of the groupoid core of the diagram category. In the case of the compact symplectic groups, i.e., automorphism groups of quaternion vector spaces, we provide a comparison with the other known versions of Weiss calculus analogous to the comparisons of calculi of the second named author, and classify certain stably trivial quaternion vector bundles over finite cell complexes in a range, using elementary results on convergence of Weiss calculi.

Symplectic Weiss calculi

TL;DR

The paper develops two symplectic Weiss calculi: a non-compact version based on symplectic vector spaces, which deformation retracts onto unitary calculus via the groupoid-core equivalence, and a compact quaternion version built with quaternionic Stiefel combinatorics that parallels and contrasts unitary and orthogonal calculi. The deformation retraction is established by comparing indexing categories through Quillen's bracket construction and showing that the unitary and symplectic calculi are equivalent at the level of ∞-categories and their Weiss towers. In the quaternionic setting, derivatives are classified by spectra with Sp(n) action, yielding a precise Homog^n(J^H,S_*) ≃ Sp^{B Sp(n)} correspondence and a robust tower formalism. Applications include the classification of stably trivial quaternionic vector bundles over finite complexes, described via stable maps to threefold suspensions of stunted quaternionic projective spaces, and a detailed comparison framework linking quaternionic calculus with unitary and orthogonal calculi. The findings illuminate the universality of Weiss calculus across symplectic geometries and provide concrete computational tools for stable quaternionic bundles and their moduli.

Abstract

We provide two candidates for symplectic Weiss calculus based on two different, but closely related, collections of groups. In the case of the non-compact symplectic groups, i.e., automorphism groups of vector spaces with symplectic forms, we show that the calculus deformation retracts onto unitary calculus as a corollary of the fact that Weiss calculus only depends on the homotopy type of the groupoid core of the diagram category. In the case of the compact symplectic groups, i.e., automorphism groups of quaternion vector spaces, we provide a comparison with the other known versions of Weiss calculus analogous to the comparisons of calculi of the second named author, and classify certain stably trivial quaternion vector bundles over finite cell complexes in a range, using elementary results on convergence of Weiss calculi.
Paper Structure (28 sections, 41 theorems, 99 equations)

This paper contains 28 sections, 41 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Symplectic Weiss calculus
  3. Weiss calculus and groupoid cores
  4. Quillen's bracket construction
  5. Groupoid cores
  6. Functors on groupoid cores
  7. Generalized polynomial functors
  8. Symplectic Weiss calculus and the deformation retraction
  9. The unitary and symplectic groupoids
  10. Symplectic Functors and polynomial approximation
  11. The deformation retract of calculi
  12. Quarternion Weiss calculus
  13. Quaternion Steifel combinatorics
  14. Quaternion vector spaces
  15. Stiefel Combinatorics over the quaternions
  16. ...and 13 more sections

Key Result

Theorem A

A version of Weiss calculus exists for functors from vector spaces with a symplectic form to spaces, and this Weiss calculus is equivalent to unitary calculus.

Theorems & Definitions (88)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Remark 2.1.1
  • proof
  • Definition 2.2.1
  • Lemma 2.2.2
  • proof
  • ...and 78 more