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      4
      votes
      0answers
      65 views

      Homotopy pullback of motivic weak equivalences

      How to see that whether $\mathbb{A}_1$-weak equivalence is closed under homotopy pullback? Let $L_{Nis}(Sm_S)$ be the Nisnevich localization, how to compute the homotopy pullback of maps $\mathbb{A}_1\...
      2
      votes
      1answer
      200 views

      Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

      Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
      7
      votes
      2answers
      264 views

      Model category structure on spectra

      I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it. Let $S$ be a finite dimensional Noetherian scheme ...
      34
      votes
      2answers
      3k views

      Why is Voevodsky's motivic homotopy theory 'the right' approach?

      Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
      4
      votes
      0answers
      153 views

      Which models are available for the motivic homotopy category $SH^{S^1}(k)$

      The motivic $S^1$-stable homotopy category $SH^{S^1}(k)$ (where $k$ is a field that is often assumed to be perfect; yet one can probably take quite general base schemes here) is "intermediate" ...
      6
      votes
      1answer
      261 views

      More on categories of modules over the algebraic cobordism spectrum

      I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning ...
      8
      votes
      0answers
      149 views

      Unaugmentable cosimplicial simplicial sheaves and realization functor

      I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...
      3
      votes
      0answers
      135 views

      Connecting Quillen functors between motivic homotopy categories (of different “types”): references?

      For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it: (a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
      15
      votes
      1answer
      431 views

      What are the advantages of various “models” for the motivic stable homotopy category

      People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
      12
      votes
      2answers
      525 views

      Simple question: different definitions of Bousfield localization

      I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let $\mathbf{...
      3
      votes
      1answer
      235 views

      Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

      I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
      3
      votes
      0answers
      161 views

      “Extending scalars” for (motivic) ring spectra and for modules over them: are the corresponding Moore spectra highly structured ring objects?

      Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a commutative ring object in any symmetric stable model category); let $R$ be an associative commutative ...
      9
      votes
      1answer
      519 views

      Is the injective model structure on symmetric spectra Bousfield localizable?

      I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...

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