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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.

        1
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        0answers
        43 views

        Straightening of a marked simplicial set

        Let $X$ be in $Set_{\Delta}$, let $\phi=id:\mathbb{C}X\rightarrow \mathbb{C}X$ be the map of simplicial categories over which we want to straighten. Assuming that $X$ is a quasi-category, how can one ...
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        votes
        1answer
        465 views

        What does the homotopy coherent nerve do to spaces of enriched functors?

        Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...
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        votes
        0answers
        72 views

        Prove that $\text{Hom}_R (I(M)/M, I(M)) = 0$? [on hold]

        Problem: Suppose $I(M)$ is an injective hull of $R-$module $M$. Prove that each isomorphism $\varphi \colon I(M) \rightarrow I(M)$ has this property $\varphi(x)=x, \forall x \in M$ is the identity ...
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        0answers
        31 views

        Semisimplicity of the tensor identity in a multifusion category over an arbitary field

        For a multifusion category $ \mathcal{C} $ over an algebraically closed field it is known that $ \text{End}(\mathcal{1}) $ is a commutative semi-simple algebra. See, for example, Theorem 4.3.1 in [1]. ...
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        votes
        2answers
        440 views

        Describing fiber products in stable $\infty$-categories

        Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
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        votes
        1answer
        166 views

        The category of complexes over a dg-algebra is Grothendieck (it has a generator)

        Let $A$ be a dg-algebra over some commutative ring $k$. We have an abelian category $\mathrm{C}(A)$ of (right) $A$-dg-modules. I've read in a few sources that $\mathrm{C}(A)$ is a Grothendieck abelian ...
        3
        votes
        1answer
        285 views

        Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

        Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
        4
        votes
        0answers
        167 views

        Very canonical constructions

        You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ a construction. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this ...
        4
        votes
        0answers
        128 views

        Compact objects in the $\infty$-category presented by a simplicial model category

        Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-...
        10
        votes
        0answers
        226 views

        When is Fun(X,C) comonadic over C with respect to the colimit functor?

        Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular,...
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        0answers
        57 views

        Composition of monads induces tensor product in the category of modules

        I have recently asked a question about the composition of two monads, namely $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_C$. I am conjecturing that the cateogory of $\mathbb{C}$-Modules and the ...
        3
        votes
        0answers
        89 views

        Quantum Scattering Experiments: C-Modules, N-Modules and Their Monads

        I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check. The cateogory of $\mathbb{C}$-Modules is monadic over set The category of $\mathbb{N}...
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        votes
        1answer
        171 views

        Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?

        A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...
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        0answers
        84 views

        Monomorphisms in the slice category

        Let $Sch$ denote the category of all schemes, and given a scheme $S$, let $Sch_S$ denote the category of $S$-schemes. Let $F$ denote the forgetful functor $Sch_S\rightarrow Sch$. Given two objects $...
        4
        votes
        0answers
        112 views

        Category of separated presheaves (over any site)

        It is well-known that the category of discrete fibrations over a category $\mathbb{C}$ is equivalent to the category of presheaves on $\mathbb{C}$. More generally I think it is true, and probably ...

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