127 questions
Filter by
Sorted by
Tagged with
52 views

• 2,081
29 views

### Biproduct in arrow category of additive category

Let $\mathcal{A}$ be an additive category and $\text{Arr}(\mathcal{A})$ be its arrow category i.e. the objects are the morphisms in $\mathcal{A}$ and the morphisms from $u:A\to B$ to $v:C\to D$ are ...
• 3,482
1 vote
62 views

• 3,757
1 vote
172 views

### Kernels and cokernels for the category of long exact sequences

I've seen in many places, including Weibel's excellent book, that the category of short exact sequences (in an abelian category) has kernels and cokernels, although it is not abelian. However, I ...
166 views

### Additive functors as colimits of representable functors

Let $A$ be an additive category. Denote $(A, Ab)$ the category of all additive functors from A to the category of abelian groups. There is well known result that any functor $F \colon C \to Sets$ on a ...
• 6,467
298 views

### Zero object in additive category

I am new in category theory and I got stuck with this following problem - An additive category $\mathcal{C}$ is a category with $\mathcal{C}(A, B)$ an abelian group for every pair $A, B$ and ...
• 382
72 views

### Maps of complexes are homotopic if and only if a certain diagram exists

$\DeclareMathOperator{\mc}{Mc}$ $\DeclareMathOperator{\diff}{Diff}$ $\DeclareMathOperator{\fct}{Fct}$ $\newcommand{\c}{\mathscr{C}}$I'm trying to understand and prove the following statement : Let $\c$...
• 1,048
1 vote
196 views

• 3,482
129 views

### Category of torsion free abelian groups is additive but not abelian

I’m trying to prove that the category of torsion free abelian groups is additive but not abelian… To prove that it is not abelian I’m trying to prove that that the multiplication map φ x2 in Z is not ...
• 21
49 views

### Additive category without finite limits

I was wondering if they are easy examples of additive categories wherein the equalizer or the pullback do not always exist ?
• 1,058
1 vote
225 views

### Canonical morphism $\text{im}(f)\to\text{ker}(g)$ for exact sequence in an abelian category

Let $\mathcal{A}$ be an abelian category. Suppose we have objects $A$, $B$ and $C$, and morphisms $f:A\to B$, $g:B\to C$ with $g\circ f=0$ i.e. is equal to the zero morphism $A\to C$. In order to make ...
• 3,482
148 views

### Understanding a product of objects as a limit of a discrete diagram

I have read that a product of objects $\{A_i\}_{i\in I}$ in a category $\mathcal{A}$ can be defined as a limit of a discrete diagram i.e. a diagram $D:\mathcal{I}\to\mathcal{A}$ where the only ...
• 3,482
1 vote
121 views

• 1,323
153 views

### Category of Abelian group pairs is not Abelian

Consider the category of pairs $(X,Y)$ where $X,Y$ are Abelian groups such that $X\subseteq Y$. I want to show that this category is not Abelian. I have checked that this category is additive, and ...
• 1,052
97 views

### In a additive tensor category, End(1) ,where 1 is a unit object, is a commutative ring

I am reading Tannakian categories by Deligne and Milne. The question is this : Let (C,#) is an additive tensor category and (1,e) is an identity object then R= End(1) which acts on each object of X, ...
• 43
87 views

### Canonical morphism is always monic and epic?

I'm reading Manin & Gelfand's book Methods of Homological Algebra and I came to something I can't prove after several tries. It's on page 113, II.5.11.(b) Commentary on the Axiom A4 (of Abelian ...
• 1,237
1 vote
109 views

### Is there a version of Short Five Lemma in any Exact Category?

It is well known that Short five Lemma (https://en.m.wikipedia.org/wiki/Short_five_lemma) holds in any abelian category. My question is: Does a version of Short five Lemma also hold in exact category? ...
• 535
1 vote