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      Questions tagged [model-categories]

      The tag has no usage guidance.

      4
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      52 views

      Cotangent complex and its distinguished triangle- a generalisation?

      Associated to any ring maps $A\to B\to C$ there is the distinguished triangle $$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\...
      4
      votes
      1answer
      166 views

      Thomason fibrant replacement and nerve of a localization

      The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...
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      votes
      3answers
      406 views

      On model categories where every object is bifibrant

      Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one ...
      6
      votes
      2answers
      247 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 ...
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      votes
      2answers
      183 views

      Example of non accessible model categories

      By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...
      2
      votes
      0answers
      81 views

      Why is a homotopy limit of a cosimplicial space not the ordinary limit?

      I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...
      7
      votes
      2answers
      404 views

      What are the advantages of simplicial model categories over non-simplicial ones?

      Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
      8
      votes
      0answers
      178 views

      $\Gamma$-sets vs $\Gamma$-spaces

      I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
      12
      votes
      1answer
      370 views

      Counter-example to the existence of left Bousfield localization of combinatorial model category

      Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists ...
      6
      votes
      1answer
      241 views

      Is the Thomason model structure the optimal realization of Grothendieck's vision?

      In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
      3
      votes
      1answer
      272 views

      Definition A.3.1.5 of Higher Topos Theory

      I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...
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      votes
      2answers
      292 views

      For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

      How explicit are the model structures for various categories of spectra? Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...
      3
      votes
      1answer
      93 views

      Monoidalness of a model category can be checked on generators

      If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating ...
      8
      votes
      2answers
      244 views

      Localization, model categories, right transfer

      Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $...
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      votes
      0answers
      45 views

      Simplicial models for mapping spaces of filtered maps

      Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan. Suppose that ${...

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