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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.

        3
        votes
        0answers
        23 views

        Induction principle from naturality

        The Church encoding of the sum type $A + B$ goes like that: $$\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X$$ But it lacks an induction principle. According to this blog article ...
        8
        votes
        2answers
        251 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 ...
        3
        votes
        0answers
        77 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 $A$. ...
        11
        votes
        1answer
        208 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 ...
        6
        votes
        1answer
        203 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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        votes
        2answers
        220 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 ...
        7
        votes
        2answers
        270 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 ...
        8
        votes
        1answer
        116 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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        votes
        0answers
        67 views

        Natural transformation of $A_\infty$-functors lifted from homology

        Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...
        3
        votes
        1answer
        90 views

        Monoidalness of a model category can be checked on generators

        If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating ...
        5
        votes
        1answer
        350 views

        Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory

        In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits. For the life of me I cannot follow the proof. Can ...
        40
        votes
        2answers
        1k views

        Ultrafilters as a double dual

        Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known: $X$ canonically embeds into $\beta X$ (by taking principal ultrafilters); If $X$ is finite, then there ...
        6
        votes
        2answers
        178 views

        When is a fold monomorphic/epimorphic

        Given a functor $F : \mathcal C \to \mathcal C$ with initial algebra $\alpha : FA \to A$, and another algebra $\xi : FX \to X$, we obtain a unique morphism $\mathsf{fold}~\xi : A \to X$ such that $\...
        8
        votes
        2answers
        236 views

        Localization, model categories, right transfer

        Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $...
        12
        votes
        2answers
        454 views

        Why is Kan's $Ex^\infty$ functor useful?

        I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...

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