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### 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 ...
• 39
34 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 ?
• 938
45 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 ...
• 2,550
47 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 ...
• 2,550
1 vote
52 views

• 1,031
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### 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 ...
• 563
36 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, ...
• 23
54 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 ...
• 1,219
1 vote
80 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? ...
• 381
1 vote
82 views

• 182
1 vote
71 views

### Is the functor that takes an $R$-algebra to the group of finitely generated projective modules of rank one additive?

Let $P$ be the covariante functor from the category of commutative $R$-algebras to the category of abelian groups that takes an $R$-algebra $T$ to the group $P(T)$ of the finitely generated projective ...
1 vote
51 views

### Isomorphism of functors in additive categories

Let $U$,$V$ be two additive categories and $F$,$G:U → V$ additive functors. If there exist natural isomorphisms in $M\in V$ and $N\in U$ $$\phi:M,N:V(M,F(N))→V(M,G(N))$$ I want to prove that there ...
• 13
1 vote
36 views

I do not see any natural way how a coslice category of a preadditive category can be preadditive (other than in some degenerate cases). However, they are given as an example in Popescu's book "...
• 1,638
129 views

### Derived functor of additive functor

Let $\mathscr{C}$ be a category which admits enough injectives and let $\mathscr{I}$ be the full subcategory of injective objects. Let $F:\mathscr{C}\rightarrow\mathscr{C}'$ be an additive functor of ...
• 367
53 views

### $K$-group of category of bounded chain complexes of free modules with finite length homologies

For a Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free modules with ...
• 3,593
82 views

### Alternating sum of length of homologies of chain complexes of modules

Let $R$ be a Commutative Noetherian ring. For a chain complex $A_*$ of finitely generated $R$-modules which is homologically finite (i.e. only finitely many homologies $H_i(A_*)$ are non-zero) and ...
• 3,593
1 vote
85 views

### Can an object be a proper direct summand of itself?

Let $\mathcal{A}$ be an additive category, $A,C$ two objects in it. If $A\oplus C\cong A$, is it true that $C=0$? It seems rather clear, but I am not finding anything on the web and can't really ...
336 views

• 5,936
169 views

### Grothendieck group of equivalent categories

Let $\Gamma : \mathcal A \to \mathcal B$ be an equivalence of categories where $\mathcal A$ is an exact category, $\mathcal B$ is an additive full subcategory of the category of $R$-modules for some ...
• 4,224
76 views

### Why is $E^*(X)$ graded commutative?

Context: I'm reading The stable homotopy category by Malkiewich (see his webpage, at Expository writings), and at this point the author hasn't actually introduced spectra, just the properties we would ...
• 41.6k
201 views

### Prove that the kernel of a homomorphism between Modules Satisfies the Universal Property

My professor gave our class this question, and the result seems simple enough. Being new to category theory I just wanted to ask if my proof looks alright. $\textbf{Proof:}$ Suppose that $R$ is a ...
• 2,648
29 views

### Examples of codim-2 objects in extended TQFT

I'm scratching my head trying to understand what an extended TQFT associates to $(n-2)$-hypersurfaces. Here's some intuition that I've developed. For an $(n-1)$-hypersurface chopped into $(n-2)$-...
89 views

### Proof explaining opposite of what is observed

Little time ago I started to playing with binary numbers and just have a little fun . I found some pretty interesting things. I want to ask if my own findings have some official names or no .Also I ...
• 325
175 views

### Category where morphism sets are Abelian groups

Let $\mathcal{C}$ be a category, and suppose for all objects $X, Y$ of $\mathcal{C}$, Mor$_{\mathcal{C}}(X,Y)$ is equipped with the structure of an Abelian group, such that the composition of ...
• 1,591
93 views

### if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
• 213
285 views

### Abelian group structure of the additive category

Let $A$ be an additive category. We have that for all $X, Y$ objects in $A, Hom_A(X, Y)$ is an abelian group. However, what is the abelian group structure? If $f, g: X \rightarrow Y$ are two morphisms ...
• 3,196
I'm trying to understand the definition of an Ab-enriched category, but I don't get how a hom-set can have a group structure. Doesn't $Hom_C(a, b)$ consist only of morphisms of the form \$f: a \...