The Morel-Voevodsky Construction over Algebraic Stacks

2025/06/15 by Neeraj Deshmukh, Deshmukh, Neeraj, Felix Sefzig +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2506.12820

openalex publication_date 2025/06/15 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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 à la Drew-Gallauer to algebraic stacks.

Citations

• Non-representable six-functor formalisms
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• A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry
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• Motivic infinite loop spaces
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• The six operations for sheaves on Artin stacks II: Adic Coefficients
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• Compactifying the space of stable maps

Related

• On the Motivic Homotopy Type of Algebraic Stacks (with an appendix joint with Jack Hall)
• Motivic homotopy theory and stable homotopy groups
• Generalized cohomology theories for algebraic stacks
• Unstable Motivic Homotopy Theory
• Motivic classifying ∞-topoi and spectral stacks