Questions tagged [additive-categories]
For questions dealing with additive or pre-additive categories.
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functors preserve isomorphism of direct sum.
In this proof in the Stacks project, it is mentioned that the decomposition of identity morphism of direct sum:
... because the composition $F(A) \oplus F(B) \xrightarrow{\varphi} F(A \oplus B) \...
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The functor that preserves the direct sum is an additive functor
I am learning about additive categories on the Stacks Project, and I encountered some difficulties in proving that functor that preserves direct sums is an additive functor.
There is such a ...
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Prove that any preadditive category can be embedded into an additive category as a full subcategory
I am trying to prove that any preadditive category $\mathcal{C}$ can be embedded as a full subcategory into an additive category $\mathcal{D}$.
Since an additive category is, by definition, a ...
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Constructing Shift Functor on Chain Complexes - Alternating Differential
I've got a question pertaining to the construction of the shift functor on the chain complex category Ch$(\mathbf{A})$ (where $\mathbf{A}$ denotes an additive category).
It makes perfect sense to me ...
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Why Do Additive Categories Need Zero Objects? - Motivation
At the moment, I'm trying to develop intuition behind the derived construction of what is an additive category from the (standard) category definition.
(i) It seems natural that a category with the ...
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In what sense are preadditive categories also enriched categories?
I'm confused about Wikipedia's definition of preadditive categories:
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that ...
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Left adjoint of an additive functor between triangulated categories that commute with shift
Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, ...
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The collection of maps to a (commutative) monoid is a (commutative) monoid, via Eckmann-Hilton
A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of ...
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Does a category where binary products commute with binary coproducts necessarily have biproducts?
Given a category $\mathcal{C}$ with binary products and binary coproducts for which the canonical comparison morphism $(A \times B) + (C \times D) \to (A+C) \times (B + D)$ is an isomorphism for any ...
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Universal property of additive categories
Keep in mind that I suck at cateogory theory, so please be gentle.
The definition of additive categeory that I have is that it is a preadditive category $\mathcal{C}$ with an object $0$ such that End$(...
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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 ...
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On the proof of the Additive Yoneda Lemma
The additive Yoneda lemma says that if you have an $Ab$-enriched category $\mathscr A$ and an additive functor $F:A\rightarrow Ab$, then there is a group isomorphism
$$Nat(\mathscr A(A,-),F)\cong F\ A$...
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How does ${f_i}^j=\pi_j\cdot f\cdot \text{in}_i$ construct matrices in additive categories?
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$ $\text{Definition 1:}$ A $\textbf{...
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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 ...
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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 ...
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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 ...
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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 $\c$...
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Weak kernels and cokernels in a triangulated category
Weak kernels and cokernels are defined to be the same as kernels and cokernels but with the requirement of uniqueness removed from their universal properties.
I want to prove that each morphism $u:A\...
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Examples of preadditive categories that is not additive
We follow the definition of stacks project. So the existence of zero object is not in the definition of preadditive categories.
Clearly any ring $R$ (commutative with 1) thought of as a category with ...
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'Invariants' in a category of modules
I have a commutative unital ring $R$, a full additive subcategory $\mathcal{C}$ of $\text{Mod}_R$ that is closed under isomorphisms and an operation $f \colon \mathrm{Ob}(\mathcal{C}) \to \mathbb{Z}_{\...
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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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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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