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Beck-Chevalley Fibrations

Thomas Holme Surlykke

Abstract

We extend the theory of ambidexterity developed by M.J. Hopkins and J. Lurie by proving commutativity of the norm square induced from a weakly ambidextrous morphism by two Beck-Chevalley fibrations that are associated by a functor. By showing how ambidexterity is preserved under base change of Beck-Chevalley fibrations, we demonstrate that our result is a generalization of the naturality property of the norm shown by M.J. Hopkins and J. Lurie. Furthermore, we demonstrate how our generalization implies two specific results previously shown by S. Carmeli, T. M. Schlank, and L. Yanovski, namely, that the induced norm square of local systems, and the induced norm square of equivariant powers, both commute.

Beck-Chevalley Fibrations

Abstract

We extend the theory of ambidexterity developed by M.J. Hopkins and J. Lurie by proving commutativity of the norm square induced from a weakly ambidextrous morphism by two Beck-Chevalley fibrations that are associated by a functor. By showing how ambidexterity is preserved under base change of Beck-Chevalley fibrations, we demonstrate that our result is a generalization of the naturality property of the norm shown by M.J. Hopkins and J. Lurie. Furthermore, we demonstrate how our generalization implies two specific results previously shown by S. Carmeli, T. M. Schlank, and L. Yanovski, namely, that the induced norm square of local systems, and the induced norm square of equivariant powers, both commute.
Paper Structure (16 sections, 26 theorems, 71 equations)

This paper contains 16 sections, 26 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Outline
  4. Conventions
  5. Acknowledgements
  6. Preliminaries
  7. The Straightening Theorem
  8. Beck-Chevalley Fibrations and Ambidexterity
  9. The Norm Square
  10. Beck-Chevalley Transformations
  11. The Norm Square Theorem
  12. Base Change of Beck-Chevalley Fibrations
  13. Applications
  14. Naturality of the Norm
  15. The Norm for Local Systems
  16. ...and 1 more sections

Key Result

Theorem 1

Let $A$ be a Kan complex. Assume that for every vertex $x \in A$, the sets $\pi_{n}(A, x)$ are finite for every integer $n$ and trivial for $n \gg 0$. Let $\rho \colon A \longrightarrow \mathop{\mathrm{Sp}}\nolimits_{K(n)}$ be a diagram of $K(n)$-local spectra, indexed by $A$. In this situation, the

Theorems & Definitions (78)

  • Theorem : Hopkins-Lurie
  • Theorem : Greenlees-Hovey-Sadofsky
  • Definition 2.1: htt
  • Remark 2.2
  • Definition 2.3: htt
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.9
  • Remark 2.10
  • ...and 68 more