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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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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1answer
76 views

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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53 views

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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1answer
40 views

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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2answers
33 views

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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1answer
40 views

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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0answers
38 views

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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15 views

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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73 views

(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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0answers
34 views

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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41 views

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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1answer
57 views

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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1answer
90 views

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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1answer
100 views

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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1answer
45 views

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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1answer
82 views

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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1answer
63 views

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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1answer
146 views

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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1answer
29 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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43 views

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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1answer
100 views

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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2answers
52 views

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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1answer
43 views

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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1answer
58 views

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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1answer
27 views

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 $ ...
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0answers
16 views

Internal additions in additive categories agree with given ones

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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1answer
54 views

Equalizers exist in an Abelian category

I'm trying to show that equalizers exist in an Abelian category. I am trying to follow a proof my professor did in class, but it's hazy. I understand we first consider the monomorphisms $(1,f),(1,g):A ...
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2answers
37 views

Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & ...
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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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1answer
23 views

Transferring Exactness

If $$\begin{matrix}0&\rightarrow&A&\rightarrow&B&\rightarrow&C&\rightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow\\ ...
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1answer
34 views

How do the first three axioms of an abelian category imply that hom-sets are enriched over the monoidal category of abelian groups? [duplicate]

According to this article in Wikipedia the following first three axioms in the definition of of an abelian category imply that hom-sets are enriched over the monoidal category of Abelian groups: ...
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2answers
70 views

Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
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1answer
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Example of epimorphisms such that the product is not an epimorphism in the category of sheaves

I've heard that in the category of sheaves over a topological space $X$, products of epimorphisms are not epimorphisms. I think that it's equivalent to saying that $\mathbf{Sh}(X)$ does not satisfy ...
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1answer
66 views

Why does the pushout preserve monic in an abelian category?

In this question the poster says that, if one of the two maps with the same domain is monic, then the corresponding induced map in the pushout diagram is also monic, in an abelian category. ...
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4answers
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Why is $\operatorname{Hom}(A, B)$ an abelian group?

Can someone please explain why a Hom-set (the set of all morphisms between two abelian groups $A$ and $B$) does also form an abelian group with addition? By the way both groups $A$ and $B$ have the ...
4
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1answer
100 views

Factoring morphisms in abelian categories

I am reading the appendix of Charles Weibel's Homological Algebra and have the following question. It is mentioned that every morphism $f: B \to C $ in an abelian category factors as $B \to im(f) \to ...
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An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are ...
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2answers
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I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work?

Wikipedia defines the notion of an abelian category as follows (link). A category is abelian iff it has a zero object, it has all binary products and binary coproducts, and it has all ...
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3answers
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If $\text{ker }f$ lies in $\mathcal{C}$ then $\text{ker }f$ is the same in $\mathcal{A}$.

I have seen this used as an argument in my textbook: Set-up: Assume $\mathcal{C}$ is a full subcategory of an Abelian category $\mathcal{A}$. Let $f: A \rightarrow B$ be a morphism in $\mathcal{C}$. ...
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2answers
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Characterisation of abelian categories in which colimit of subobjects are subobjects

This question is related to 1 and 2. Given an abelian category $\mathcal{C}$ in which colimit exists. What is a necessary and sufficient condition on $\mathcal{C}$ so that given any $X \in ...
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1answer
57 views

A property of abelian categories

My aim is to show that the category of free abelian groups not an abelian category. I read that I could fix $n \in \mathbb{N} \setminus \left\lbrace 0,1 \right\rbrace $ and consider the ...
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1answer
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Chinese remainder theorem in abelian categories

The chinese remainder theorem holds in arbitrary abelian categories? I found a generalization in homological categories, but i'm looking for a proof in valid in an abelian category.
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1answer
47 views

Choice of the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. PS: I previously posted a similar question which didn't make a lot of ...
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Choosing the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. Suppose we have two exact sequences $$ 0\to A \to B \to C \to D \to E ...
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1answer
57 views

If we have finite coproducts in Ab-enriched category, does it follow that the zero object exists?

I am trying to understand what exactly are the axioms for an additive category and I got a bit lost. If we are given a category that is Ab-enriched and admits finite coproducts, can we derive it that ...
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2answers
79 views

How do we get the canonical cokernel-kernel decomposition in a pre-abelian category?

In a pre-abelian category, every morphism $f: A \to B$ has a canonical decomposition: $$ A \to coker(kerf) \to ker(cokerf) \to B $$ How do we obtain the middle morphism, the one from $coker(kerf)$ ...
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1answer
59 views

Pullback preserves cokernel

Is that true that in an abelian category $\mathcal{C}$ if I have the pullback diagram: $$ \require{AMScd} \begin{CD} P @>{p_1}>> C\\ @V{p_2}VV @V{g}VV \\ A @>{f}>> B \end{CD} $$ ...
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1answer
73 views

Prove an isomorphism via abstract nonsense

Suppose we are working in an abelian category and we have a commutative diagram with exact rows $$ \newcommand{\ra}[1]{\kern-1em\xrightarrow{#1}\kern-1em} ...