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Questions tagged [additive-categories]

For questions dealing with additive or pre-additive categories.

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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
56 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
45 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
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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
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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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1answer
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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
67 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
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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
42 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
72 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
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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
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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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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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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 (skeletally ...
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1answer
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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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1answer
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Locally finitely presented category

In page 1 of "Locally finitely presented additive categories", author says that a locally finitely presented category $\mathcal{A}$ is one for which every object can be expressed as a direct limit of ...
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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
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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
43 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
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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
112 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
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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
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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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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
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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
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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 ...
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1answer
106 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
47 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
83 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 ...
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0answers
85 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 $F$ ...
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3answers
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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 $M = 0$. ...
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1answer
368 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 ...
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2answers
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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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1answer
100 views

On the additive category

On the book,categories for working mathematician, by S.Mac Lane, he gave a definition as follows: an additive category is an Ab-category which has a zero object 0 and a biproduct for each pair of its ...
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0answers
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Dold Kan correspondence for additive categories

Let $\mathcal{A}$ be an additive category satisfying the following property. $$(\ast)\ \text{For every }i\colon A\rightarrow B\ \text{in }\mathcal{A}\text{ which has a left inverse, there is an ...
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1answer
119 views

Zero object in a preadditive category

Let $\mathcal{A}$ be a category such that for each $A,B\in \mathcal{A}$ the Hom-set $\operatorname{Hom}(A,B)$ is given the structure of an abelian group. We require that the composition law for ...
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1answer
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Universal property of images in category theory

Let $\mathcal{A}$ be an additive category with all kernels and cokernels and $f:A\to B$ a morphism. If $e:B\to \text{coker}(f)$ is the canonical epimorphism, define $\text{im}(f):=\ker(e)$, with a ...
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0answers
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Proving End_A (A') for additive category A' has ring structure

I am currently trying to prove that if $A$ is an additive category and $A'$ is an object, then $End_A (A')$ has a natural ring structure. The only part I am not sure about is proving the distributive ...
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2answers
108 views

Single-object additive category

A pre-additive category with a single object $\bullet$ is simply a ring $R = \mathrm{Hom}(\bullet,\bullet)$: pre-additivity makes this Hom-space an abelian group and with bilinear composition, i.e. a ...
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2answers
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The completeness of a category of additive functors between additive categories

In what follows $\textbf{preadditive}$: a category $\mathscr{C}$ is preadditive when $\forall\ A,B,\ \mathscr{C}(A,B)$ is an abelian group and the morphisms composition is a group homomorphism on ...
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2answers
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Morphisms between biproducts in additive categories

I can't understand the following description of a morphism between biproducts in an additive category, which I found in Borceux, Vol.2 If $A_1,A_2,B_1,B_2$ are four objects in an additive category $\...
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2answers
373 views

Zero objects in preadditive categories

Let $\mathscr{C}$ be a preadditive category. Let $A,B$ be two objects and assume that $\mathscr{C}$ has a zero object $0$ (an object both initial and terminal). Let $f$ be the unique zero morphism ...
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1answer
582 views

Why is the category of finitely generated modules over a non-noetherian ring not abelian?

I am learning about abelian categories for a talk I have to give next week. One of the first questions I had upon learning this definition is "does there exist an additive category that is not abelian?...
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1answer
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Can tensor abelian categories always be embedded into the category of modules?

Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such ...
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1answer
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Are pullbacks of regular epimorphisms in an additive category always epic?

Fix a additive category $\mathsf C$ (that is to say, $\mathsf C$ admits an additive structure and has a zero object all biproducts). Given four morphisms in $\mathsf C$ as follows: $$\require{AMScd} ...
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Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
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1answer
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Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff $(0\...
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1answer
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Increasing value, decreasing amount

So I really only came here for one question, since I have no idea what it's called. Let's use a comparison to make it a bit easier. I have two coins of the value '1'. When I add those together, I ...
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2answers
487 views

Flat Modules are Filtered Colimits of Free Modules

A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ...
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1answer
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Write formulas in specific languages of group.

So, for each of the following groups write a formula in the language of group theory, which holds in given group, but doesn't hold in others two. $(i)$ The integers with addition \ I think it's $\{\...