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        Questions tagged [model-categories]

        The tag has no usage guidance.

        7
        votes
        0answers
        151 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
        317 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
        216 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
        253 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 ...
        7
        votes
        2answers
        274 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
        91 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
        240 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 $...
        2
        votes
        0answers
        44 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 ${...
        3
        votes
        1answer
        59 views

        Simplicial models for fibrations between mapping spaces

        Let $S,T$ 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. We let $|S|$ denote the geometric ...
        4
        votes
        1answer
        183 views

        simplicial objects in a model category

        Suppose that we have a (combinatorial if necessary) model category $M$, and let $F:\Delta^{op}\rightarrow M$ a simplicial object in $M$, such that for any natural number $n$, $F([n])$ is a fibrant ...
        6
        votes
        1answer
        125 views

        Simplicial localization of the cofibrant-fibrant objects

        Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...
        3
        votes
        1answer
        155 views

        Cofibrations of functors

        Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...
        3
        votes
        0answers
        114 views

        Why is the Straightening functor the analogue of the Grothendieck construction?

        In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...
        6
        votes
        1answer
        272 views

        Model structure on the category of topological groups

        Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...
        11
        votes
        1answer
        222 views

        Which maps of simplicial sets geometrically realize to fibrations?

        If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...

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