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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A necessary and sufficient condition for contravariant auto-equivalence on module categories

I have a problem about the condition of contravariant auto-equivalence on module categories. Let $R$ be a algebra over a field. Let $\mathcal{C}$ be a abelian subcategory of $R$-modules, and assume ...
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Common kernel for compositions of epis?

The following proposition is an excerpt from Osborne's *Basic Homological Algebra: Regarding the proof: Why does there exist an arrow $j$ which is simultaneously the kernel of both $\pi$ and ...
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Abelian categories with tensor product

Is there a standard notion in the literature of abelian category with tensor product? The definition ought to be wide enough to encompass all the usual examples of abelian categories with standard ...
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$\operatorname{Im}f\cong A/\operatorname{Ker}f$ in abelian categories

Let $f:A \rightarrow B$ be an arrow in some abelian category. There is the usual epi-mono factorization of any such arrow, but can we go further and prove isomorphism of the objects: ...
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Projection between quotients by related subobjects

For a subobject $A\overset{a}{\rightarrowtail} B$ we define the quotient object $B\twoheadrightarrow B/A$ as the cokernel of any monic representing the subobject. Suppose we have another subobject ...
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Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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Borceux - Snake Lemma Question

Below is the statement of the snake lemma from Borceux. My question is which squares are (1) and (2) referring to?
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Quotient objects as constructions from subobjects?

A quotient object of an object $A$ is usually denoted $A/B$ (we're talking about equivalence classes of epis). It seems that in categories like $\mathsf {Grp}$ and $\mathsf {Ab}$ one can associate ...
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Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
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Finding suitable basis for a free abelian finitely generated group.

I am stuck with this exercise forever... I was barely taught about it, English is not my mother language and in any other phrasing it is not coherent with my material.I'd really appreciate your help. ...
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103 views

Query in the definion of abelian category

I am studying the definition of abelian category..Definition says it is a additive category with a)every morphism in category has kernel and co-kernel. b)every monomorphism in the category is the ...
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Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
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Extension of nonisomorphic simple objects

Let $X$ and $Y$ be two nonisomorphic simple objects in an abelian category. Are all extensions of $X$ by $Y$ trivial? ( $\mathrm{Ext}^1(X,Y)=0$ ?)
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A Lemma from Freyd

This is a lemma from Freyd's Abelian Categories stated without proof. In an abelian category, $$A\rightarrow S \rightarrowtail B = A \rightarrow B$$ if and only if $$A\rightarrow B \twoheadrightarrow ...
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Freyd: “is a subobject of” is not transitive

On page 20 of Abelian Categories, Freyd writes Note that the relation "is a subobject of" is not transitive. On page 91 of Awodey's Category Theory (there are several typos in this page; the ...
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Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$?

Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$? What about $\mathsf{Grp}$ makes for a seemingly far-more-complicated coproduct? If your answer revolves around ...
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59 views

If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?

I didn't understand a step in the proof of Proposition 5.92 from Rotman's Introduction to Homological Algebra (2nd Ed.) where he says: "there is a morphism $g: B\to C$ [in a given abelian category ...
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96 views

Which categories of linear representations are semisimple?

Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) ...
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Quotient Object of Subobjects

A problem (not homework) from CWM: For subobjects $u\leq v$ of an object $a$ in an abelian category $\mathsf A$, define a "quotient" object $v/u$ (to agree with the usual notion in $\mathsf ...
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Meaning of $f=me$ Factorization in Abelian Categories

Propsition 1, part 1 (Maclane, CWM p.199) Let $\mathsf A$ be an abelian category. Then every arrow has a factorization $f=me$, with $m$ monic and $e$ epic; moreover, $$m=\ker (\text{coker} ...
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Fibre products and induced short exact sequences in abelian categories

Assume we have an abelian category which has fibre products. Let $f:X\to Z$ and $g:Y\to Z$ be two morphisms and let $(W,p,q)$ be their fibre product with $p:W\to X$, $q:W\to Y$. If the category is a ...
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Uniqueness of the long exact sequence in homology

A few days ago colleagues of mine and I listened to a talk about spectral sequences and one "application" of them was the proof that any short exact sequence (s.e.s.) $$0 \to A \xrightarrow{f} B ...
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Homotopy of double chain complexes

Consider complexes $(A,d_1), (A',d_1)$, $(C,d_2), (C',d_2)$ and morphisms $f_1,f_2: (A,d_1)\to (A',d_1)$ and $g_1,g_2: (C,d_2)\to (C',d_2)$ of degrees $0$. Consider the functor $(-\otimes-)$, then ...
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Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ ...
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Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
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Converse of the Nine-Lemma (aka $ 3\times 3$ lemma)

I have been asked to either prove or disprove a sort of converse to the well know "Nine Lemma" (Also sometimes called the $3 \times 3$ Lemma I believe) The basic concept of the Nine Lemma is that if ...
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Spectral sequence of a filtered complex: convergence conditions and abelian categories

There is a theorem that if given a filtered complex and the filtration is bounded then there is a spectral sequence whose 0th and 1st page have specific forms and the sequence converges to ...
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Compatibility of homomorphisms and quotient maps of abelian groups

Suppose $A$ and $C$ are abelian groups with subgroups $A'$ and $C'$ respectively. Let $f:A\to C$ be a group homomorphism. I was wondering if the following statements are equivalent: There exists a ...
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Mitchell's Embedding Theorem for not-necessarily-small categories

Mitchell's Embedding Theorem states that if $\mathcal{A}$ is a small abelian category, then there is a ring $R$ and a fully-faithful exact functor $F:\mathcal{A}\rightarrow R\mathsf{Mod}$. To what ...
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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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Quasi-isomorphisms are localizing in the homotopy category of cochain complexes

I'm having trouble grokking the proof of the above fact in Gelfand-Manin, Theorem 4 of III.4, page 161-162. I don't think it makes sense to copy out everything here, I'll just assume you have a copy. ...
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(Are there) subtleties in the definition of 'biproduct'

I always thought that a biproduct of two objects $A_1,A_2$ in some category $\mathcal{C}$ is an object $P$ with two maps $p_i:P\to A_i$ making it a product and two maps $j_i:A_i\to P$ making it a ...
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Axioms of Abelian Category [duplicate]

I know that the one of the axioms of abelian categories is that the induced morphism $ \text{coker}(\ker f ) \longrightarrow \ker ( \text{coker} f ) $ for any morphism $ f $ is an isomorphism. ...
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The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...
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In an abelian category,every morphism can be written as composition of epi and mono. [duplicate]

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
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Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
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Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
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Two definitions of homology

Let $f,g$ be arrows in an abelian category such that the composite $gf$ is defined and is given by the zero arrow. I shall try to find a definition for the quotient $\ker g /\operatorname{im} f$, ...
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Quasi-isomorphisms not localizing in Kom(A)

I've been reading up on the construction of derived categories. I understand why we prefer localizing with respect to a localizing class of morphisms (to get a nice representation of morphisms as ...
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Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ ...
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Repetitive Algebra.

I am studying the category of finitely generated left modules over the repetitive algebra and I'm using the book of Happel: Triangulated categories in the representation theory of finite dimensional ...
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showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
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33 views

Extending morphism between derived functors

Let N,M be objects in a left exact functor $F:A\rightarrow B$ between abelian categoire's source, most importantly say there is an isomorpihsm $\psi: F(M)\rightarrow R^dF(N)$ is it possible to extend ...
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exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
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Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
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What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
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The intersection of $u_{A} : A \longrightarrow A + B$ and $u_{B} : B \longrightarrow A+B$ is zero.

I am trying to show that the intersection of $u_{A}:A \longrightarrow A+B$ and $u_{B}:B \longrightarrow A+B$ is the zero map. Here, the $u_{A}$ and $u_{B}$ are the embedding maps into the coproduct of ...
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What is the cokernel of $\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$?

These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} ...
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morphism from coproduct into product in abelian categories

Let consider an abelian category with a projective generator $ R $. The canonical morphism $\coprod R\to\prod R $ (taken over the same indexing set $ I $) is monic? The answer is affirmative if $ I $ ...