Linked Questions

5
votes
0answers
360 views

kernel of cokernel is cokernel of kernel [duplicate]

Possible Duplicate: Equivalent conditions for a preabelian category to be abelian Let $\mathcal{C}$ be an abelian category, and consider an arrow $f:A\rightarrow B$. In a number of sources (Vakil'...
0
votes
0answers
105 views

Image in abelian categories [duplicate]

$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$For a morphism $ f: A\to B$ in an abelian category, we let $\im f:=\ker(\coker f)$. Then the morphism $A\to \im f$ is an epimorphism and $\...
1
vote
0answers
60 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. Let'...
0
votes
0answers
58 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 ...
20
votes
1answer
3k views

In an additive category, why is finite products the same as finite coproducts?

In an additive category, why is finite products the same as finite coproducts? This is relatively easy to prove when the category is R-mod, but my intuition/creativity fails to see how the method can ...
0
votes
1answer
311 views

$A \longrightarrow \text{Im} f$ is an epimorphism and a cokernel of ker $f \longrightarrow A$ in every abelian category

How can I verify the following statement? The image of a morphism $f : A \longrightarrow B$ is defined as $\text{Im}(f) = \text{Ke}r(\text{coke}r f)$ whenever it exists (e.g., in every abelian ...
1
vote
1answer
242 views

$\operatorname{Im}f\cong A/\operatorname{Ker}f$ 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: $\operatorname{Im}...
3
votes
1answer
219 views

Equivalent definitions of abelian categories - reference request

Etingof et. al. define abelian categories as additive categories in which for every morphism $\phi : X \to Y$ there exists a sequence $$K \xrightarrow{k} X \xrightarrow{i} I \xrightarrow{j} Y \...
2
votes
1answer
176 views

On pushouts and mapping cylinders in exact categories

Let $\mathcal N$ be an exact category and $C\mathcal N$ be the category of chain complexes with its usual exact structure. We have here the usual notion of "mapping cylinder" of a chain map. If $f:N\...
2
votes
1answer
58 views

Converse to: Equivalent conditions for a preabelian category to be abelian

In the following question: Equivalent conditions for a preabelian category to be abelian. How is the converse easily shown? I see why every monomorphism, f, is the kernel of the cokernel of f, but why ...
3
votes
1answer
92 views

Induced Exact Sequences in Abelian Categories

Let $\mathcal{A}$ be an abelian category. Show that for every $f:A\to B$ the following sequences are exact: $$0\to \text{ker}(f)\xrightarrow{i} A\xrightarrow{\pi}\text{coim}(f)\to 0$$ $$0\to \...
1
vote
1answer
67 views

$\overline{f}$ is isomorphism in abelian category

Suppose $f: A \longrightarrow B$ is a morphism in an abelian category $\mathcal{C}$. What I consider an abelian category: $\mathcal{C}$ is additive. Every morphism has a kernel and a cokernel. Every ...