Tagged Questions

Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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Why homotopy category is not abelian?

Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are ...
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Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
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Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all $\overline{\mathbb{Q}_{\ell}}$-...
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A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal A$...
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An abelian category such that all objects are injective

The problem is 'Let C be an abelian category such that all objects in C are injective. Prove that all abjects are projective.' If C has enough projectives, then the 'Ext' functor can be defined. Thus ...
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Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...
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Finitely generated abelian groups form an abelian subcategory of $\mathbb{Z}$-Mod

According to Weibel's Homological Algebra book a subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ is called an abelian subcategory if it is abelian and an exact sequence in $\mathcal{B}$ ...
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Exponential objects of internal objects respecting evaluation (2-exponentials?)

Let $(F,+_F)$, and $(G,+_G)$ be two commutative internal monoids in Sets. Set being cartesian closed, I can form $G^F$ as a set. My question is simple: is there a canonical/universal way to enforce ...
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Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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Noetherian objects are stable by extension

Let $\mathcal{A}$ be an abelian category. An object $M$ in $\mathcal{A}$ is noetherian if any ascending chain of subobjects of $M$ is stationary. (In analogy with modules.) I am trying to prove that ...
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Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
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Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...
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Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram below,...
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How to prove that a particular (sub-)category has a projective generator.

Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional. How does one ...
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I know that if $\mathcal C$ is a semiadditive category, then for every two objects $A$, $B$, the set $\mathrm{Hom}_\mathcal{C} (A, B)$ is automatically endowed with a structure of commutative monoid, ...
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Duality in abelian categories.

Let $C$ be an abelian category, with a projective separator $k$. Assume that $C$ has a duality, that's a functor $\ast:C\to C^{\text{op}}$ together with a natural isomorphism $\tau:1\to \ast\ast$ such ...
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A functor which preserves short exact sequence also preserves long exact sequence?

Let $F: C \to D$ be a functor between abelian categories (e.g. modules over the same ring), and it preserves short exact sequence, then is it also preserves long exact sequence?
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Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which 1) all kernels and cokernels exist 2) all monos are the kernel of their cokernel, all epis are the cokernel of ...
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Is $\mathbb{AB}$ an additive (or even abelian) category?

Notation. Let $U_0 \in U_1 \in U_2$ be Grothendieck universes, each containing $\mathbb N$. Let $\mathbf{Cat}_{U_0}$ be the ($U_1$-small) 2-category of all $U_0$-small categories, $\mathbf{Cat}_{U_1}$...
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Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
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Any straightforward proof of “in an abelian category, a pullback yields a monomorphism at cokernel level”?

Here is the question I encountered: \require{AMScd} \begin{CD} s @>{f^\prime}>> a @>{\varphi^\prime}>> \bar a\\ @V{g^\prime}VV @V{g}VV @V{\bar g}VV\\ b @>{f}>> c @>{...
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When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
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Some propreties about $\mathfrak{Coh}_X$ and $\mathfrak{QCoh}_X$.

I would like to know : why is the category $\mathfrak{Coh}_X$ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $\mathfrak{QCoh}_X$ of quasi ...
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Category of Morphisms Between Modules

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules. We define the category ...
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$E^{\bullet} \rightarrow \text{cone}^{\bullet}(u)[-1]$ is a quasi isomorphism

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $E^\bullet$ be a cochain complex with objects in $\mathcal{A}.$ Let $i^n : E^n \hookrightarrow I^n$. Put $F^n = I^n \oplus I^{n+1}$,...
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