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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Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
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Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
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Direct sums of semisimple objects

Let $\mathcal{A}$ be an abelian category. Call an object $M\in\mathcal{A}$ semisimple if every exact sequence $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$ splits. Is it ...
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Subobject of a direct sum in terms of components?

Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ ...
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Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
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Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
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Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
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Five Lemma with category theory

I know how to prove the 5 Lemma by diagram chasing. I would be interested in seeing a proof which only uses category theory. Does anybody know some reference where this is done?
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Proving that $(Tor_n(\_\ ,N))_n$ is a universal homological $\delta$ functor

Problem: Let $N$ be a left $R$-module, for some ring $R$. Let $T_n$ denote $Tor^R_n(\_\ , N)$. Let $(S_n)$ be another homological delta-functor from $mod$-$R$ to $Ab$, with a natural transformation ...
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Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
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Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
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Coproducts and products are same in any preadditive category

Here is the proof that coproducts and products are same in any preadditive category from the Stack project. I have few questions regarding the above proof. I don't understand what do they mean by ...
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Does $\cdots \to G_1\overset f\to G_2 \overset g\to G_3\to \cdots$ exact imply $0\to \ker(g) \to G_2 \to \operatorname{coker}(f)\to 0$ exact?

Given a (part of a) long exact sequence of abelian groups (or modules over some commutative ring) $$ \cdots \to G_1\overset f\to G_2 \overset g\to G_3 \to \cdots $$ we have the short exact sequence $$ ...
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Splitting lemma for many (at least 3) components

I am interested in such version of splitting lemma: So given short exact sequence $\hskip2.5in$ we have three equivalent statements: short exact sequence is right split, i.e there is map $t: ...
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Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
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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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Why is Grp not an Abelian Category?

As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the ...
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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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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 ...
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Wiki on exact sequences in regular categories

In a regular category, an exact sequence is a diagram which is both a coequalizer and a kernel pair: $$R\overset r{\underset s\rightrightarrows} X\to Y$$ Wiki says that in the abelian case, the above ...
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$P\cong P^\ast$ iff $P$ is a f.g projective module?

Is it true that for a noncommutative $R$, a module $P$ is f.g projective iff $\mathsf{hom}(P,R)=P^\ast \cong P$? Here's what I thought of as a proof: Since $(-)^\ast$ is additive, it preserves ...
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“Every equivalence or duality of abelian categories is exact”

In the wiki entry on exact functors, it is written that "every equivalence or duality of abelian categories is exact". The second example given is of the dual of a vector space. Now I'm pretty sure ...
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Sincere module and Grothendieck Group

Let $A$ be basic finite-dimensional $K$-algebra and $K$ algebraically closed. Let $F$ be the free abelian group generated by representatives of the isomorphism classes of objects in $mod A$. We ...
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Tensor products and morphisms

Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$. Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$. Let $m\in \mathbb{N}$ and let $g:mX_j \to ...
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Short Exact Sequence from Single Arrow in Abelian Category

Work in an abelian category. I'm aware that given an exact sequence, one can break it into short exact sequences like so: I was wondering whether it was possible to do derive one of these short ...
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Projective cover and epimorphism

Let $C$ be an abelian category and let $X$ be an object with finite length. Thus there is a composition series $0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} ...
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finite length object and direct sum in an abelian category

Let $C$ be an abelian category and let $X$ an object with finite length. Then $X$ has a composition series $$0=X_0<X_1< \cdots X_n=X$$ where $X_i/X_{i-1}$ is simple for $i=1, \dots, n$. ...
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Epimorphisms in an abelian category

Let $C$ be an abelian category. Let $f:X \to Y$ and $g: Y \to Z$ are morphisms of $C$ and suppose that $g\circ f$ is epimorphism. Question Is $g$ epi as well? If so, I want to know the proof.
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Projectivity and semisimple in an abelian category

Let $C$ be a locally finite $k$-linear abelian category. Here locally finite means that the home sets are finite dimensional vector space and every object of $C$ has finite length. Suppose that each ...
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Is there a projection to a subobject of an object in an abelian category

Let $C$ be an abelian category (maybe, plus some nice properties.) Let $X$ be an object of $C$ and let $i:Y\to X$ be a (representative of) subobject of$X$. Question 1: Is there a projection $p$ from ...
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Surjectiveness of convolution

Consider the convolution mapping $j^*: Hom(X, Y) \otimes X \to Y$, given by bilinear formula $(\phi, x) \mapsto \phi(x)$, in a category of coherent sheaves or, generally, in any abelian category. I ...
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Serre quotient category

Let $\mathcal{A}$ be an abelian category. A Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$A \longrightarrow ...
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Infinite product of a short exact sequence

I was trying to show that in an abelian category satisfying (AB4)* the product of a short exact sequence is a short exact sequence. Given $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i ...
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What is the necessary and sufficient condition for abelian categories to have arbitrary direct limit?

As a beginner to learn homological algebra, I have just learned about the direct system and its direct limit.As R-mod categories have arbitrary coproducts indexed ...
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Are there right-deformations for abelian sheaves?

A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). ...
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Functorial injective embeddings in Grothendieck categories

I'm trying to read section 11 of the Stacks Project pdf on injectives, but I can't penetrate what's going on at all behind the proof of Theorem 11.6, which says Grothendieck categories have functorial ...
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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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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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Let $\mathscr{A}$ be a chain complex. Show that the kernel of the map $A_n/B_n \rightarrow Z_{n-1} $ is isomorphic to $H_{n}(A)$.

Let $A_{n+1} \xrightarrow{p_{n+1}} B_n \xrightarrow{r_{n+1}} Z_n \xrightarrow{k_{n+1}} A_n$, where $k_{n+1} \circ r_{n+1}=i_{n+1}$ is the monomorphism in the image factorization of $d_{n+1}$ the ...
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Is a kernel in a full additive subcategory also a kernel in the ambient abelian category?

Setting: Let $\mathscr{C}\subset \mathscr{A}$ be a full additive subcategory of an abelian category. Let $C,C'$ be objects of $\mathscr{C}$ and let $f\in ...
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Show that homology is a functor in a pure categorical way.

Let $\mathscr{A}$ be an abelian category i want to show that $\mathcal{H^i}$ ( the i-th homology group) is a functor from the category of complexes of $\mathscr{A}$ to $\mathscr{A}$. I showed this for ...
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Infinite complexes as functor categories.

I was thinking that if we have have an infinite complex in an abelian category $\mathscr{C}$ : $...\xrightarrow{f_{i-1}} \mathcal{A}^i \xrightarrow{f_{i+i}}\mathcal{A_{i}}...$ Can be regarded as a ...
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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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Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence $ {}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of ...
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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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Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
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Examples of preadditive categories

The obvious examples for preadditive categories / $\textsf{Ab}$-enriched categories are of course: $R$-$\textsf{Mod}$, the category of $R$-modules for any ring $R$ $\textsf{Ab}$, the category of ...
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Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...