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For questions dealing with additive or pre-additive categories.

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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 ...
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0answers
50 views

There is only one additive structure on a category

The book "Handbook of Categorical Algebra Volume 2. Categories and Structures" by Francis Borceux states that On a category $C$, any two additive structures are necessarily isomorphic. But ...
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1answer
41 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 ...
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30 views

Localization of an additive category by certain subclass of morphism is still additive.

I started to study localization in additive categories via a subclass of morphism which satisfies the Ore conditions by my own. Right now, Im studying how for an additive category $C$ the ...
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1answer
28 views

Are coslice categories of preadditive categories preadditive?

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 "...
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31 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 ...
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0answers
43 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 ...
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0answers
34 views

Zero element of $\mathrm{Hom}_C(X,Y)$ in the additive category $C$ [duplicate]

Let's assume $C$ is an additive category. How can we show that for each $X,Y\in\mathrm{Ob}(C)$, the morphism $X\longrightarrow 0\longrightarrow Y$ is the zero element of $\mathrm{Hom}_C(X,Y)$?
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44 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 ...
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3answers
57 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 ...
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3answers
172 views

Examples of additive categories

There are a lot of interesting and creative examples of categories, such as for example, the category whose objects are the positive integers and the set of morphisms from $n$ to $m$ is the set of $m \...
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2answers
46 views

Semantic doubt about enrichment of categories and what it means to have 'group structure'

We define a category $C$ to be $Grp$-Enriched if for every $X$,$Y\in C$, we have that $C(X,Y)$ 'has group structure'/is a group. But what does that mean really? If I am given any finite set $A$, there ...
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1answer
51 views

Mapping cone is unique up to (non-unique) isomorphism

I am reading about triangulated categories. According to the axioms, given a morphism $f : A \to B$, there is a so-called mapping cone $C$ of $f$, i.e., an object $C$ with maps $g: B \to C$ and $h: C \...
4
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1answer
113 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 ...
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1answer
48 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 ...
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27 views

Product $(P, p_A, p_B)$ exists in preadditive category $\implies \exists s_A$ such that $p_A \circ s_A = 1_A, p_B \circ s_A = 0$

This is in Borceux's Handbook of Categorical Algebra Vol II, page 5. In the start of the proof it says: Define $s_A : A \to P$ as the unique morphism with the properties $p_A \circ s_A = 1_A, \ ...
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1answer
106 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 ...
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1answer
24 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 $(n-2)$-...
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1answer
81 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 ...
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1answer
88 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 ...
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1answer
61 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 ...
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2answers
135 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 ...
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2answers
376 views

How can a hom-set have a group structure?

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 \...
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1answer
92 views

Direct Sum of Additive Categories

I am following Etingof et al's book on tensor categories. They define the direct sum of additive categories as follows (I am rephrasing): Let $\{ \mathcal C_\alpha, \alpha \in I \}$ be a family of ...
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1answer
120 views

diagonal and codiagonal morphism in additive category

Let $\mathcal{C}$ be an additive category. I define diagonal and codiagonal morphisms as follows: Diagonal: Let $A \in Ob(\mathcal{C})$. Then the diagonal $\Delta_A: A \longrightarrow A \oplus A$ is ...
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1answer
58 views

A different definition of coproduct in a category

I am following Pavel et al book "Tensor Categories". They write that an additive category is a category $\mathcal{C}$ such that: My problem is with (A3). i know that what they are trying to say is ...
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1answer
83 views

additive functor preserving byproducts preserves finite products

Let $F:C \rightarrow D$ be a functor between two additive categories. The claim (p53, prop 40) is : If $F$ preserves biproducts then $F$ preserves finite products. For the proof the author wrote:...
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1answer
92 views

Who for the first time defined abelian categories?

Who for the first time defined additive categories? Who for the first time defined abelian categories? I am guessing it should be in an algebraic geometric paper, but who and when? Any reference will ...
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1answer
45 views

Constructing limits in an additive category given the existence of products and kernels

The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
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2answers
89 views

Why is a direct summand of a compact object compact?

In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism $$ \coprod_{i} \text{Hom} \left( A, X_{i} \right)...
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1answer
90 views

Karoubian Category

One calls a category $\mathcal{C}$ karoubian if it is additive and for $X$ every idempotent map $e: X \to X$ (idempotent means $e = e^2$) splits, therefore there exist $Y$, $p:X\to Y$ and $q:Y \to X$ ...
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185 views

extending functors

In "Functors on locally finitely presented additive categories", by H. Krause, one can read (in page 108): Proposition 2.3. There is, up to equivalence, a bijective correspondence between (...
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1answer
125 views

How to think about the octohedral axiom for triangulated categories?

I am currently learning about triangulated categories and came across the following diagram in Gelfand & Manin: Here the triangles with a star inside are distinguished while the triangles with ...
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0answers
51 views

axioms of additive categories

One of the axioms of additive categories is the following: (Axiom A3) For any pair of objects $X_1,X_2$ there exist an object $Y$ and morphisms $p_1,p_2,i_1,i_2$ \begin{equation*}X_1\xrightarrow{...
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1answer
118 views

Identity making each $\operatorname{Hom}_\mathcal{A}(A,B)$ an abelian group in an additive category the Zero map?

In an additive category, we have a zero object and every Hom-set is given the structure of an abelian group such that composition distributes over addition. That is, given $f \in \operatorname{Hom}_\...
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1answer
45 views

Submodules in a preadditive category with one object.

I'm trying to establish the full link between this generalization of rings on preadditive categories and the basic case where one considers a category $\mathcal{A} = \{X\}$ with one object. Is it easy ...
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0answers
147 views

Interpretation of preadditive categories as rings.

My motivation comes from How to see that a one-object pre-additive category is a ring? and this master thesis. Basically, I want to see concrete examples of preadditive categories and what structure ...
4
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1answer
181 views

Proving the direct sum of two distinguished triangles is a distinguished triangle

Suppose that we are working in a triangulated category and there are two distinguished triangles $X_i\longrightarrow Y_i\longrightarrow Z_i\longrightarrow X_i[1]$ ($i=1,2$). I am stuck in proving that ...
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1answer
40 views

In an additive category, why a morphism between complexes can be factored as a composition of a homotopy equivalence and a monomorphism?

I was stuck in the following Problem. Suppose that $\mathcal A$ is an additive category and $f\colon X\to Y$ is a morphism of (cochain) complexes in $\mathcal A$. Then there is a factorization $f=...
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1answer
180 views

Additive Yoneda Lemma

I'm studying Abelian Categories in F. Borceux "Handbook of Categorical Algebra" Vol.2. In this reference we can find an additive version of the famous Yoneda Lemma : Let $\mathcal{C}$ be a ...
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0answers
85 views

Mapping cone and additive functors

Let me set the notations: $(\mathcal{A}, T)$ will be an additive category with translation $T(X)$ will denote the application of the functor $T$ to an object $X \in \mathcal{A}$ A differential ...
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1answer
42 views

Is the kernel of projections in an additive category preserved by additive functors?

Let $\mathcal{C}$ and $\mathcal{D}$ be an additive categories and let $\psi : \mathcal{C} \to \mathcal{D}$ be an additive functor. I have two questions : Is it true that if $f: C \to C$ is a ...
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2answers
358 views

Definitions of an additive functor

In my Rings and Modules class, we defined additive functors this way: A (covariant/contravariant) functor $F$ is called additive if: (i). $F0 = 0$, where $0$ is the zero-module. (ii). For any ...
2
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1answer
149 views

Difference between monoidal and tensor categories

Is a monoidal category just another word for tensor category or are those two different (but still similiar) things in the sense that one of them is more general? Are those categories supposed to be ...
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1answer
86 views

Additive functors preserve direct summands

My question is simple - I just can't seem to prove it for the life of me. Let $$0\rightarrow L \rightarrow M \rightarrow N\rightarrow 0$$ be a split short exact sequence of $R$-modules. Let also $F:R-...
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0answers
144 views

Question about how to get image/coimage factorization in a preabelian category

Let $\mathcal{C}$ be an additive category and assume that all finite limits and colimits exist. I am using the definition of the image and coimage of a morphism $f: A \longrightarrow B$ given here. I ...
2
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0answers
115 views

A canonical natural transformation $H\to G$ when $G \dashv F \dashv H$ and $F$ is fully faithful.

Let $F: \mathcal{A}\to \mathcal{B}$ and $G,H: \mathcal{B} \to \mathcal{A}$ be additive functors between additive categories so that we have adjunctions $$ G \dashv F \dashv H $$ and suppose that $F$ ...
2
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3answers
147 views

Hom set in Additive Category is $0$

I know this is a really easy question but for some reason I'm having trouble with it. If $M$ is an object in an additive category $\mathcal C$, and $\text{Hom}_{\mathcal C}(M,M) = 0$, then $M = 0$. ...
2
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1answer
670 views

Commutative ring category is not additive category

I was told that commutative ring category is not additive category all the time and it is thus not abelian category. A category is additive if $Hom(A,B)$ is abelian group for all objects $A,B$ and ...
5
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2answers
298 views

pullback is preserved by left exact covariant functor $T : \mathsf{Mod}_R \to \mathsf{Ab}$

This is a problem in Rotman Homological Algebra, 5.16. The following is the statement of the problem. Prove every left exact covariant functor $T:\mathsf{Mod}_R \to \mathsf{Ab}$ preserves ...

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