The Morel-Voevodsky Construction over Algebraic Stacks

TL;DR

This paper extends the motivic homotopy category construction to algebraic stacks, demonstrating equivalences with existing categories and applying it to spectral rigidity, framed categories, and coefficient systems.

Contribution

It introduces a new construction of the motivic homotopy category for algebraic stacks and proves its equivalence with existing frameworks, extending several key theorems.

Findings

Construction of the motivic homotopy category over algebraic stacks

Equivalence with the stable motivic homotopy category of Chowdhury and D'Angelo

Extension of spectral rigidity and reconstruction theorems to algebraic stacks

Abstract

In this article, we give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by Chowdhury and D'Angelo. As an application, we extend Bachmann's spectral rigidity theorem to algebraic stacks. Moreover, we extend the construction of the framed motivic homotopy category to algebraic stacks and prove Hoyois' Reconstruction Theorem in this setting. Finally, we discuss an extension of the formalism of cocomplete coefficient systems \`a la Drew-Gallauer to algebraic stacks.