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      Questions tagged [ct.category-theory]

      Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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      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 ...
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      33 views

      Terminology: a certain semicategory with objects mor(C) (not the usual or twisted arrow category)

      I’ve had recent cause to consider the following construction: given a category $\newcommand{\C}{\mathbf{C}}\C$, define a semicategory $M(\C)$, whose objects are arrows of $\C$, and where a map from $f ...
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      47 views

      Does this elementary generalization (left and right sided products) of categorical product have an official name?

      The question and definition at hand are related to the fact that in freshman calculus, to check pointwise continuity of $f:R \rightarrow R$, we ask if both the left and right hand limits exist, and if ...
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      84 views

      Sketch of sketches, or sketch of presentations

      In Sketches: Outline with References 4.3, Wells cites the result that sketches are sketchable by a finite limit sketch. I can't find the Burroni 1970a paper, and I am having a lot of trouble with Lair ...
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      250 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 ...
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      54 views

      Reference request : Isomorphic stacks are given by Morita equivalent Lie groupoids

      Let $\mathcal{G},\mathcal{H}$ be Lie groupoids. Let $B\mathcal{G}$ denote the stack of principal $\mathcal{G}$ bundles and $B\mathcal{H}$ denote the stack of principal $\mathcal{H}$ bundles. Then, we ...
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      1answer
      189 views

      Can groups be recovered as “monoids” in a bicategory?

      Is there a bicategory $V$ and a definition of monoid in a bicategory so that $\text{Monoids}(V)$ is the category of groups and homomorphisms? EDIT: For example, is there a bicategory $V$ so that ...
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      368 views

      What are the advantages of simplicial model categories over non-simplicial ones?

      Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
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      395 views

      Examples of transfinite towers

      I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops ...
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      115 views

      Orthogonality and 2-filtered 2-categories

      Let $C$ be a category. It is well known that $C$ is $\omega$-filtered if and only if it is weakly right orthogonal to every $A\to A^\rhd$, where $A^\rhd$ is the right cone of a finite category $A$. ...
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      222 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 ...
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      220 views

      Is the Thomason model structure the optimal realization of Grothendieck's vision?

      In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
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      2answers
      261 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 ...
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      278 views

      For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

      How explicit are the model structures for various categories of spectra? Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...
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      1answer
      122 views

      Weighted (co)limits as adjunctions

      It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...

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