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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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.

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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1answer
73 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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61 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
99 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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28 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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42 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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85 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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51 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
42 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
48 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
24 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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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
41 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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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
22 views

Transferring Exactness

If $$\begin{matrix}0&\rightarrow&A&\rightarrow&B&\rightarrow&C&\rightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow\\ ...
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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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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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51 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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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 ...
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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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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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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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Caracterisation 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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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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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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45 views

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
55 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
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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
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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
66 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} ...
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The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
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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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Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
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A question about pre-additive category

Let $C$ be a pre-additive category with a zero object $O$. Suppose that every morphism in $C$ has a kernel and a cokernel and that every monomorphism in $C$ is a kernel of some morphism. Prove that ...
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Some questions on abelian category

Let $f: C \longrightarrow D$ be a morphism in an abelian category $\mathfrak{A}$ with kernel and cokernel both zero. How can I show that it is an isomorphism? I am not able to find it's inverse. ...
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1answer
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How to prove exactness implies complex?

In an abelian category, there are notions of exact sequence and complex. Since the objects there may not be abelian groups, the definition of exact sequence and complex are all complicated. And the ...
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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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Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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1answer
79 views

Definition of the image as coker of ker == ker of coker?

The standard categorical definition of image is that it is the cokernel of the kernel. Under what nice conditions does this definition coincide with kernel of the cokernel? It coincides for abelian ...
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Exactness of functors as “iff”; conjecture about bifunctors

The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ ...
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Does the adjugant of additive functors between abelian categories preserve the abelian structure of the hom-set?

I think the following is a counter-example. I noticed it when trying to prove that the sheafification functor induces isomorphism on the stalks (Vakil 2.4M). As in Vakil (2.6.3) the stalk functor is ...