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        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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        If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

        I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
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        194 views

        Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

        In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
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        votes
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        108 views

        How to visualize the Microsupport of a Sheaf?

        I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
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        votes
        1answer
        162 views

        What is the definition of a $\mathcal{U}$-category?

        Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category. There are two definitions of $\mathcal{U}$ category I've met. $...
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        votes
        1answer
        340 views

        Endomorphisms in the derived category

        (Apologies if this question is trivial, but I'm way outside my area here.) Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived ...
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        80 views

        DG functors along which contractions can be lifted

        For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...
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        1answer
        230 views

        Understanding the functoriality of group homology

        EDIT: I've decided to rephrase my question in order for it to be more concise and to the point. Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...
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        107 views

        Quantum groups at $q=-1$

        For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
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        1answer
        113 views

        In a fibration, how do properties of arrows downstairs affect the base-change functors?

        Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base. What are the relations between the following pairs? $$f\text{ epi}\qquad f^\ast \text{ faithful}$$ $$f\text{ mono}\qquad f^...
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        142 views

        Testing for equivalences of $\infty$-categories on strictifications?

        It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences. Question : Can we do something similar for: quasi-categorical ...
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        3answers
        200 views

        Constructive proof of exponential law in a category

        I'm trying to write a constructive proof of the isomorphism ${Z}^{X \times Y} \equiv (Z^Y)^X$ in a category with exponential objects. I can construct a map $(Z^Y)^X \rightarrow {Z}^{X \times Y}$, but ...
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        0answers
        144 views

        Tannaka duality for $DG$ indschemes

        In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism $$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$ where $X$ and ...
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        0answers
        407 views

        Turning a category into a semigroup

        Consider a small (that is objects and morphisms belonging to a Grothendieck universe) category $\mathcal{C}$, whose objects are all small sets, and an identity-on-objects functor ${\uparrow}: \mathbf{...
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        1answer
        62 views

        sequences of iterated orthogonals (lifting property) in a category

        I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property. For example, several iterated orthogonals of $ \emptyset\...
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        1answer
        78 views

        Equivalent conditions to be a lax idempotent contravariant monads

        $\require{AMScd}$As already said, a contravariant monad is "like a monad, but a contravariant functor": the multiplication and unit are dinatural transformations an algebra is a map $a : TA\to A$ ...

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