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      Questions tagged [model-categories]

      A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.

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      80 views

      Existence of tensor product of infinity operads

      I am trying to show, or find a reference, for the following fact: "Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product". In other ...
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      votes
      0answers
      53 views

      On cofibrations of simplicially enriched categories

      Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells. We have a canonical inclusion functor , $$i: C \...
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      votes
      1answer
      185 views

      Compact objects in the $\infty$-category presented by a simplicial model category

      Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-...
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      votes
      0answers
      78 views

      Homotopy colimits of simplicial objects

      Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...
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      votes
      0answers
      66 views

      Right adjoint preserving (trivial) cofibrations between cofibrant objects

      Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects. ...
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      78 views

      Which set of compact objects generates the subcategory of a compactly generated stable model category?

      I couldn't find any info on what set of compact objects generates the following subcategory: Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
      2
      votes
      1answer
      79 views

      Is the Hurewicz model category left proper?

      A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak ...
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      votes
      1answer
      253 views

      Homotopy limit over a diagram of nullhomotopic maps

      Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit $$ \underset{i \in I}{\mathrm{holim}}X(i), $$ where the maps $X(i) \to X(j)$ are ...
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      votes
      2answers
      263 views

      Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

      I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...
      7
      votes
      0answers
      142 views

      Why does the cotangent complex really have a distinguished triangle?

      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
      176 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
      415 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 ...
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      votes
      2answers
      258 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
      191 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 ...
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      votes
      0answers
      84 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 ...

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