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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Homotopy category of chain complexes as a localization

For an abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the ...
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The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
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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 ...
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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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Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...
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28 views

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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83 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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What is the pullback of a central extension?

Suppose we have three objects $A,B,C$ of an (abelian) category $\mathbf{C}$ and a short exact sequence $ 0\to A \to B \to C \to 0 $ such that $B$ is a central extension of $C$ by $A$ ($im(A\to ...
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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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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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84 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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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? ...