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      6
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      The model category structure on $\mathbf{TMon}$

      I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer. I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
      3
      votes
      1answer
      135 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 ...
      2
      votes
      1answer
      200 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 ...
      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 \...
      3
      votes
      1answer
      291 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 ...
      3
      votes
      0answers
      136 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
      245 views

      Does the Dwyer-Kan model structure make dgCat a model $2$-category?

      Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category. Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
      6
      votes
      2answers
      645 views

      Theorem 2.1.2.2 Higher Topos Theory

      At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
      5
      votes
      1answer
      127 views

      Inductive folk model structure on strict ω-categories

      There is a paper of Lafont, Metayer, and Worytkiewicz [1] that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure ...
      8
      votes
      2answers
      298 views

      Are cofibrations accessible?

      The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations? More generally, let $C$ be a locally presentable ...
      12
      votes
      1answer
      288 views

      Is there an “injective version” of the Bergner model structure?

      The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions. ...
      3
      votes
      0answers
      54 views

      Calculating the intersection of the saturations of a decreasing sequence of morphisms

      I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...
      15
      votes
      3answers
      573 views

      Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?

      Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences. Let $\mathbf Q$ be the corresponding $\infty$-...
      13
      votes
      1answer
      309 views

      Weak complicial sets: Are the morphisms too strict?

      In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
      14
      votes
      2answers
      473 views

      What is a model category from an $\infty$ point of view?

      A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example: Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense ...

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