scholarly journals On the rational motivic homotopy category

2021 ◽  
Vol 8 ◽  
pp. 533-583
Author(s):  
Frédéric Déglise ◽  
Jean Fasel ◽  
Fangzhou Jin ◽  
Adeel A. Khan
Keyword(s):  
Homotopy Category ◽  
Motivic Homotopy

Geometrical symmetric powers in the motivic homotopy category

2019 ◽  
Vol 223 (12) ◽  
pp. 5396-5408
Author(s):  
Joe Palacios
Keyword(s):  
Homotopy Category ◽  
Symmetric Powers ◽  
Motivic Homotopy

scholarly journals Slices of motivic Landweber spectra

2010 ◽  
Vol 9 (1) ◽  
pp. 103-117 ◽  
Author(s):  
Markus Spitzweck
Keyword(s):  
Compact Objects ◽  
Homotopy Category ◽  
Motivic Homotopy

AbstractIn this paper we show that a conjecture of Voevodsky about the slices of the motivic cobordism spectrum implies a statement about the slices of motivic Landweber spectra. Over perfect fields these slices are given by the coefficients of the corresponding topological Landweber spectrum and the motivic Eilenberg MacLane spectrum. We also prove a cohomological version of Landweber exactness which applies to the compact objects of the stable motivic homotopy category.

scholarly journals Norms in motivic homotopy theory

Astérisque ◽  
2021 ◽  
Vol 425 ◽  
Author(s):  
Tom BACHMANN ◽  
Marc HOYOIS
Keyword(s):  
Homotopy Theory ◽  
Galois Theory ◽  
Homotopy Category ◽  
Motivic Cohomology ◽  
Power Operations ◽  
Motivic Homotopy Theory ◽  
Slice Filtration ◽  
Motivic Homotopy ◽  
Algebraic Cobordism ◽  
Spectrum Structure

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a  normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

scholarly journals Enriched simplicial presheaves and the motivic homotopy category

2011 ◽  
Vol 215 (7) ◽  
pp. 1663-1668
Author(s):  
Philip Herrmann ◽  
Florian Strunk
Keyword(s):  
Homotopy Category ◽  
Simplicial Presheaves ◽  
Motivic Homotopy

scholarly journals Brown representability in 1-homotopy theory

2011 ◽  
Vol 7 (3) ◽  
pp. 527-539 ◽  
Author(s):  
Niko Naumann ◽  
Markus Spitzweck
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Noetherian Scheme ◽  
Finite Dimensional ◽  
Brown Representability ◽  
Motivic Homotopy

AbstractWe prove the following result announced by V. Voevodsky. If S is a finite dimensional noetherian scheme such that S = ∪αSpec(Rα) for countable rings Rα, then the stable motivic homotopy category over S satisfies Brown representability.

scholarly journals Remarks on motivic homotopy theory over algebraically closed fields

2010 ◽  
Vol 7 (1) ◽  
pp. 55-89 ◽  
Author(s):  
Po Hu ◽  
Igor Kriz ◽  
Kyle Ormsby
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Motivic Homotopy Theory ◽  
Stable Homotopy Category ◽  
Motivic Homotopy ◽  
Algebraically Closed Fields

AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

scholarly journals The structure of motivic homotopy groups

2016 ◽  
Vol 23 (1) ◽  
pp. 389-397 ◽  
Author(s):  
Bogdan Gheorghe ◽  
Daniel C. Isaksen
Keyword(s):  
Homotopy Groups ◽  
Motivic Homotopy
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