Beck-Chevalley Fibrations

TL;DR

This paper generalizes the naturality of the norm in ambidexterity theory using Beck-Chevalley fibrations, showing how certain induced norm squares commute and are preserved under base change, extending prior results.

Contribution

It introduces a new framework using Beck-Chevalley fibrations to generalize the naturality and commutativity properties of the norm in ambidexterity theory.

Findings

Norm squares induced from weakly ambidextrous morphisms commute.

Ambidexterity is preserved under base change of Beck-Chevalley fibrations.

Generalizes previous results on local systems and equivariant powers.

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.