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        Questions tagged [model-categories]

        The tag has no usage guidance.

        4
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        0answers
        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
        167 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 ...
        12
        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
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        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
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        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
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        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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        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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