Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes

Abstract

Let $\V$ be a symmetric monoidal model category and let $X$ be an object in $\V$. From this we can construct a new symmetric monoidal model category $Sp^Σ(\V,X)$ of symmetric spectra objects in $\V$ with respect to $X$, together with a left Quillen monoidal map $\V\to Sp^Σ(\V,X)$ sending $X$ to an invertible object. In this paper we use the recent developments in the subject of Higher Algebra to understand the nature of this construction. Every symmetric monoidal model category has an underlying symmetric monoidal $(\infty ,1)$-category and the first notion should be understood as a mere "presentation" of the second. Our main result is the characterization of the underlying symmetric monoidal $\infty$-category of $Sp^Σ(\V,X)$, by means of a universal property inside the world of symmetric monoidal $(\infty ,1)$-categories. In the process we also describe the link between the construction of ordinary spectra and the one of symmetric spectra. As a corollary, we obtain a precise universal characterization for the motivic stable homotopy theory of schemes with its symmetric monoidal structure. This characterization trivializes the problem of finding motivic monoidal realizations and opens the way to compare the motivic theory of schemes with other motivic theories. As an application we provide a new approach to the theory of noncommutative motives by constructing a stable motivic homotopy theory for the noncommutative spaces of Kontsevich. For that we introduce an analogue for the Nisnevich topology in the noncommutative setting. Our universal property for the classical theory for schemes provides a canonical monoidal map towards these new noncommutative motives and allows us to compare the two theories.

Third version, 139 pages, Updated with some technical tools necessary for the second part of the project (see Noncommutative Motives II)

References in corpus (4)

• Higher Topos Theory
• Dendroidal sets and simplicial operads
• A guided tour through the garden of noncommutative motives
• Motivic structures in non-commutative geometry

Cited by in corpus (12)

• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• Derived Algebraic Geometry and Deformation Quantization
• Operations on derived moduli spaces of branes
• Tannaka duality and stable infinity-categories
• Model structure on projective systems of $C^{\ast }$-algebras and bivariant homology theories
• The Beilinson regulator is a map of ring spectra
• Overconvergent global analytic geometry
• The tangent bundle of a model category
• Mixed motives and quotient stacks: Abelian varieties
• Motivic classifying $\infty$-topoi and spectral stacks
• Etale homotopy theory of non-archimedean analytic spaces
• The infinity-category of stabilized Liouville sectors