Questions tagged [additive-categories]

For questions dealing with additive or pre-additive categories.

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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 ...
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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 ?
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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 ...
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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 ...
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Canonical morphism from coproduct to product in a pointed category

Suppose we have a pointed category $\mathcal{A}$ and a collection $\{A_i\}_{i\in I}$ of objects which have both a product $\prod_iA_i$ and a coproduct $\coprod_iA_i$. The product has projections $\...
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Direct sum and short exact sequence, as well as, tensor product and what?

In an abelian category, the notion of direct sum is generalized by the notion of short exact sequence (see split exact sequence). Question: In a monoidal category, can the notion of tensor product be ...
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A functor that preserves binary coproducts or binary products between additive categories must preserve the zero object?

That is the question. If I have a functor $F:\mathcal{A}\to\mathcal{B}$ between additive categories such that either $F(A)\oplus F(B)\to F(A\oplus B)$ is an isomorphism for all $A,B\in\mathcal{A}$, ...
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Canonical group structure on hom-sets in an additive category

I am trying to understand why a given category (without any extra structure) is either abelian or not abelian i.e. why the definition can be expressed in such a way that there is no need to add any ...
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Can $\mathbf{Rng}$ be made into a pre-additive category?

Question: Can $\mathbf{Rng}$ be made into a pre-additive category? Rngs are just rings, without the requirement of an identity. Accordingly, we do not require rng homomorphisms to preserve the ...
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Shift/translation functor in a triangulated category

I'm trying to get to grips with triangulated categories at the moment. According to Wikipedia, the shift/translation functor of a triangulated category $\mathcal{C}$ is an "additive automorphism (...
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Additive and preadditive categories

I have been attempting to make sense additive categories recently. I outline my current understanding below and would be very grateful if you could let me know if I have made a mistake, since I am not ...
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Alternate Definition of Quotient Category

A quotient category is usually (at least from most sources I had seen) defined by something like $\mathcal C/\sim$ for some equivalence relation $\sim$, for example in this post. I am wondering if one ...
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Prove that additive functor preserves products and coproducts

Let $\cal A,B$ be additive categories and $F:\cal A\rightarrow B$ be an additive functor. Show that $F$ preserves products and coproducts. Since product and coproduct of a pair $A,B$ of objects in an ...
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$p_Ai_B=0$ and $p_Bi_A=0$ in additive category

Let $\cal A$ be an additive category. Then for any $A,B\in\textrm{Ob}(\cal A)$ the direct sum $A\oplus B$ is both their product and their coproduct. Let $i_A:A\rightarrow A\oplus B$ and $i_B:B\...
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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 ...
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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, ...
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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 ...
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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? ...
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Weibel Lemma 1.6.2.

The following is (a part of) Lemma 1.6.2. from Weibel's Homological Algebra.$\newcommand{\C}[1]{\mathcal{#1}}\DeclareMathOperator{\coker}{coker}\newcommand{\md}[1]{{\left\lvert #1 \right\lvert}}\...
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Existence of sum/intersection of abelian (resp. additive) categories

Let $\mathcal{C}$ and $\mathcal{D}$ be abelian (resp. additive) categories. Does there exist a smallest abelian (resp. additive) category $\mathcal{C}+\mathcal{D}$, s.t. $\mathcal{C}$ and $\mathcal{D}$...
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What is the internal hom functor for vector spaces?

I am wondering what the explicit definition of the internal hom functor $[X,\_]$ (described in this answer), for the category of vector spaces $\text{Vect}$ (over, say, the field of real numbers) is. ...
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3 votes
2 answers
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Image of a morphism in a category with zero object, equalizers and coequalizers

Let’s assume that $\mathcal{C}$ is a category which has a zero object and equalizer and coequalizer of all the morphisms exist (so we have kernels and cokernels). Then, is it true that for an ...
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On a special kind of $R$-linear functor from an $R$-linear additive category [closed]

Let $R$ be a Commutative Noetherian ring, let $\mod (R)$ be the category of finitely generated $R$-modules. Let $\mathcal C \subseteq \mod(R)$ be an $R$-linear (https://stacks.math.columbia.edu/tag/...
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Two rings with the same multiplicative structure but non-isomorphic underlying Abelian groups

I am giving a series of lectures where I introduce some undergraduates to basic ideas from category theory. One of the things I would like to show them is how category theory could be used to make ...
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Question about split idempotent in triangulated categories

I've read about split idempotents, and I know they are morphisms $e:X\to X$ such that there exists $Y$ and morphisms $i:Y\to X$, $p:X\to Y$ with $pi=\mathrm{id}_Y$, $ip=e$. In the setting of ...
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Equivalence of $R$-linear additive categories are $R$-linear?

Let $R$ be a commutative ring. Let $C,D$ be $R$-linear categories (https://stacks.math.columbia.edu/tag/09MJ ) such that they are also additive categories. Let $F: C \to D$ be an equivalence of ...
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X is a direct summand of $X\otimes DX\otimes X$ in a rigid tensor-triangulated category.

I am reading Balmer's "Supports and filtrations in algebraic geometry and modular representation theory". In Prop. 2.4 he claims that $X$ (an object of a rigid tensor triangulated category) ...
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$\mathrm{Hom}$-sets in categories with finite biproducts have the structure of a commutative monoid: a reference for a proof

It seems it is a folk result that $\mathrm{Hom}$-sets in categories with finite biproducts (including the empty biproduct) have the structure of a commutative monoid. I know an analogous result for ...
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Additive closure of a category

Given a category $\mathcal{C}$, there is a (I believe) well-known way to obtain an additive category from that, called the additive closure of $\mathcal{C}$ (see eg Bar-Natan's Khovanov’s homology for ...
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A functor $\mathcal{F}$ which is not exact but it has an exact power $\mathcal{F}^n$

Is there a functor $\mathcal{F}$ on an abelian category $\mathcal{C}$ which is not exact but there is a natural number $n$ such that $\mathcal{F}^n$ is an exact functor? What about the same question ...
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Does idempotent completion commute with direct sum completion?

Definitions: For a pre-additive category $\mathcal{C}$ I denote its idempotent completion by $\overline{\mathcal{C}}^p$. The objects of $\overline{\mathcal{C}}^p$ are pairs $(X,p)$, where $X \in \...
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1 answer
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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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1 vote
1 answer
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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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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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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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2 votes
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$K$-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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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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1 vote
3 answers
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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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5 votes
3 answers
336 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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4 votes
2 answers
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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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1 vote
1 answer
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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 \...
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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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2 votes
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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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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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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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3 votes
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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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4 votes
1 answer
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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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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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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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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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