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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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        3
        votes
        1answer
        111 views

        Proving a Kan-like condition for functors to model categories?

        I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
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        votes
        2answers
        433 views

        Non-small objects in categories

        An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits. Is there an example of a (locally small) ...
        2
        votes
        1answer
        186 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 ...
        5
        votes
        0answers
        99 views

        When do zigzags of weak equivalences detect isomorphisms in the localization?

        The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question. ...
        3
        votes
        1answer
        117 views

        Strøm model structure on nonnegatively graded chain complexes

        Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes. The ...
        4
        votes
        1answer
        107 views

        Bousfield localization of a left proper accessible model category

        What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)
        3
        votes
        0answers
        164 views

        The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

        Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g. N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from https://neil-strickland.staff.shef.ac.uk/courses/...
        6
        votes
        1answer
        243 views

        Homotopy pullbacks and pushouts in stable model categories

        There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
        1
        vote
        0answers
        90 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 ...
        3
        votes
        0answers
        58 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 \...
        5
        votes
        1answer
        194 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 $\infty$-...
        2
        votes
        0answers
        87 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 ...
        4
        votes
        0answers
        83 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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        0answers
        84 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
        84 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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