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      Questions tagged [locally-presentable-categories]

      4
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      1answer
      98 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 ...
      12
      votes
      2answers
      185 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
      294 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
      234 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
      295 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
      113 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
      225 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. ...
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      votes
      2answers
      365 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
      220 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
      185 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
      407 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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      votes
      0answers
      114 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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      votes
      2answers
      116 views

      Is a filtered colimit of complete module complete?

      This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...
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      votes
      0answers
      89 views

      What additional property does the antipode give on the category of all modules over an Hopf algebra?

      It is well known that many constructions involving bialgebras extends to monoidal categories, and often becomes more natural in that framework. If one cares about the category of finite dimensional ...
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      votes
      2answers
      284 views

      A locally presentable locally cartesian closed category that is not a quasitopos

      This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...

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