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      Questions tagged [locally-presentable-categories]

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      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) ...
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      1answer
      266 views

      Is the 2-сategory of groupoids locally presentable?

      I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete. It has been pointed out that the category of groupoids is ...
      4
      votes
      1answer
      128 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)
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      votes
      1answer
      104 views

      Why is the category of all small $\mathbf{S}$-enriched categories locally presentable?

      In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$ with all objects cofibrant and weak ...
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      2answers
      200 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 ...
      11
      votes
      1answer
      305 views

      Is there any references on the tensor product of presentable (1-)categories?

      Is there any references on the tensor product of (locally) presentable categories ? All I know about this is Lurie's book that deals with the $\infty$-categorical version, and a few references that ...
      11
      votes
      1answer
      250 views

      Accessible functors not preserving lots of presentable objects

      Let $F:\cal C\to D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large ...
      14
      votes
      2answers
      343 views

      $\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects

      Remark 1.30 of Adámek and Rosicky, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be ...
      5
      votes
      1answer
      124 views

      Rank of presentability of internal Hom of locally presentable categories

      Let $C$ and $D$ be locally $\kappa$-presentable categories. It is written on the nLab that the category $\mathrm{Ladj}(C, D)$ of cocontinuous functors from $C$ to $D$ is again locally $\kappa$-...
      9
      votes
      1answer
      233 views

      Closure of presentable objects under finite limits

      In a locally presentable category $\cal E$, there are arbitrarily large regular cardinals $\lambda$ such that the $\lambda$-presentable (a.k.a. $\lambda$-compact) objects are closed under pullbacks. ...
      16
      votes
      2answers
      380 views

      Raising the index of accessibility

      In the standard reference books Locally presentable and accessible categories (Adamek-Rosicky, Theorem 2.11) and Accessible categories (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\...
      6
      votes
      1answer
      230 views

      Locally presentable categories, universes, and Vopenka's principle

      Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such ...
      3
      votes
      1answer
      191 views

      Locally presentable categories

      Under category Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...
      15
      votes
      1answer
      414 views

      When is the category of models of a limit theory a topos?

      If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory. Is there a characterization of ...
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      0answers
      127 views

      Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

      Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...

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