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      Questions tagged [accessible-categories]

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      9
      votes
      2answers
      493 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) ...
      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)
      13
      votes
      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 ...
      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 $\...
      14
      votes
      2answers
      244 views

      Is every accessible category well-powered?

      Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter. Every accessible category ...
      7
      votes
      0answers
      121 views

      Stability of accessible $\infty$-categories under some operations

      I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
      2
      votes
      2answers
      124 views

      Example: Accessible category without colimits

      I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch. Bonus points if the sketch and/or the colimit ...
      5
      votes
      0answers
      122 views

      When is $Ind(C)$ small?

      Let $C$ be a small category. Then $Ind(C)$ is the free completion of $C$ under filtered colimits. My sense is that typically, $Ind(C)$ is a large category. But sometimes it is small. For example, if $...
      11
      votes
      2answers
      479 views

      What are the reflective subcategories of the category of presentable categories?

      I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
      5
      votes
      1answer
      149 views

      Can I check the accessibility of a functor on directed colimits of presentable objects?

      Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed. Is it true that $F$ is $\lambda$-accessible if and only if ...
      7
      votes
      1answer
      176 views

      Saturated classes, generation by a set and pullbacks of categories

      Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors ...
      8
      votes
      4answers
      695 views

      What was Burroni's sketch for topological spaces?

      In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
      5
      votes
      1answer
      84 views

      About small $\omega$-orthogonality classes and Gabriel-Ulmer duality

      I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\...
      4
      votes
      1answer
      158 views

      About small-orthogonality classes of a locally presentable category

      Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable categories. $\mathcal{A}$ reflective and closed under filtered colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let $...
      4
      votes
      3answers
      248 views

      About the Yoneda objects of a locally presentable category

      This question is a follow-up of Extending functors defined on dense subcategories. Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a Yoneda object if ...

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