# Questions tagged [abelian-categories]

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185
questions

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### Small abelian categories and module categories - preservation of injective and projective objects

A soft question on small abelian categories:
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper
Wikipedia: "The article "Sur quelques points d'algèbre homologique" by ...

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176 views

### Bounds on homological dimension of functor categories

Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying ...

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### How can one characterize categories of exact functors?

Does there exist any intrinsic characterization of additive categories equivalent to $\operatorname{Ex}(A,Ab)$, that is, of exact functors from a small abelian category $A$ into abelian groups? Any ...

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208 views

### Filling square to push-out in abelian category

Let $\mathcal{C}$ be an abelian category. In $\mathcal{C}$ we consider the diagram
\begin{array}{ccc}
A&&\\\
\downarrow&&\\\
C&\rightarrow&D
\end{array}
with arrows being ...

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117 views

### Methods for finding complex for subobjects of homology

Let $\mathcal{C}$ be an abelian category and
$$ C_\bullet:C_n \rightarrow C_{n-1}\rightarrow \ldots \rightarrow C_1\rightarrow C_0$$
a complex in $\mathcal{C}$. Suppose we have for each $i$ a ...

**6**

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421 views

### Functorial kernel in derived category

By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...

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173 views

### How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?

I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...

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130 views

### Derived category of an abelian monoidal category

For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...

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393 views

### When is the category of finitely presented modules abelian?

Let $R$ be an associative ring with identity and $\mathrm{mod}R$ be the category of finitely presented $R$-modules. I would like to know when the category $\mathrm{mod}R$ is abelian.
I know that if $R$...

**19**

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**1**answer

374 views

### Vopěnka's principle and contravariant full embeddings between module categories

I was recently reminded about this old question on math.stackexchange.
Let $\operatorname{Mod}R$ be the category of (right) modules for a ring $R$. The questioner mistakenly thought that the Freyd-...

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51 views

### Connecting homomorphism and Baer sum in an abelian category

I would like to prove that the connecting homomorphism $\delta \colon \mathrm{Hom}_{\mathcal{A}}(N,M_3) \to \mathrm{Ext}_{\mathcal{A}}(N,M_1)$ from part (2) of Lemma 12.6.4 of the Stacks Project is ...

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108 views

### Subclasses of abelian categories, that are closed under extensions and their complement as well and a construction of torsion pairs using them

Let $\mathcal{A}$ be a length abelian category. A subclass $\mathcal{S}$ of $\mathcal{A}$ is called super-closed if $0\in \mathcal{S}$, $\mathcal{S}$ is closed under extensions and $\mathcal{A}\...

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102 views

### Is the category of Yetter-Drinfeld modules abelian?

Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...

**6**

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587 views

### Categorical presentation of direct sums of vector spaces, versus tensor products

My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a ...

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149 views

### Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one

I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...

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191 views

### Which abelian categories possess an exact faithful functor into abelian groups that respects products?

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ ...

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313 views

### From Topoi to Grothendieck categories

This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...

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212 views

### Is every middle exact functor a derived functor?

Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...

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106 views

### Functors with adjoints

I want to find a functor between abelian categories, which is faithful but not full. And this functor has left and right adjoint. I want to know a nontrivial example,which is not inducecd by a ring ...

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votes

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190 views

### Existence of eigenvalues in a k-linear abelian category

I cannot find any categorical definition of an eigenvalue, so I ask this question. Let $\mathbb{k}$ a be a field and $\mathcal{C}$ be a $\mathbb{k}$-linear abelian category. Let $f: X \rightarrow X \...

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134 views

### Grothendieck axioms and sheaf categories

An abelian category, $A$, is said to satisfy axiom AB6 if for every family of filtered diagrams $I_j$ indexed by a set $J$, the canonical map $\mathrm{colim}_{\prod_{I_j}}(\prod_j M_{ij}) \to \prod_J \...

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227 views

### Do the isomorphism classes of indecomposable objects in $R{\text{-mod}}$ form a set?

Let $R$ be a unital (associative) ring. Consider the category $R\text{-mod}$ of unitary left $R$-modules. Set $\text{Indec}(R)$ to be the class of all isomorphism classes of indecomposable objects ...

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35 views

### Eilenberg–Zilber-type theorem for Map([n],A), where the degeneracy maps for [n] are forgotten

The following statement should be immediately implied by Eilenberg–Zilber theorem if the sequences $(i_0,\ldots,i_k)$ below are only monotone. But I need the strict monotone version which I believe to ...

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77 views

### additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...

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402 views

### Category of modules over an Azumaya algebra and the Brauer group

Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$.
I would like to know ...

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vote

**1**answer

67 views

### Has the covariant Hom-functor of the category of additive categories a left adjoint?

Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...

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605 views

### Why do some tricks in homological algebra work over the category of C*-algebras?

The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...

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168 views

### Semisimple Abelian categories with infinite sums

A semisimple category is an abelian category in which every object is a finite direct sum of simple objects.
A) Why does one impose the finiteness condition here?
B) If one condsiders infinite direct ...

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88 views

### When uniquely divisible objects can be embedded into ind-torsion ones?

Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid_M$ is invertible. We will say that $M'$ is ind-...

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276 views

### Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor ...

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136 views

### Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian?
[stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)]
In ...

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61 views

### Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...

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votes

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157 views

### Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...

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227 views

### Freyd-Mitchell for $k$-linear categories

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of ...

**2**

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**1**answer

112 views

### Condition for an additive functor to be an equivalence

Consider an additive functor $F : \mathcal{A} \longrightarrow \mathcal{B}$ between abelian categories and suppose that $F$ is a dense functor, that is, for every object $B$ in $\mathcal{B}$ there is ...

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440 views

### Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf
sub-algebra of $H_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same ...

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98 views

### Baer sum and endomorphisms

I work in an Abelian category. If I take the Baer sum $M' + M''$ of two extensions $M'$ and $M''$ of $
M_2$ by $M_1$, i.e.,
$$ 0 \to M_1 \to M' \to M_2 \to 0$$
is exact, and the same for $M''$, then ...

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votes

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603 views

### Abelian category with enough injectives but not functorially

Let $\mathcal{A}$ be an Abelian category with enough injectives. Is it always possible to make the injective embedding functorial? By this I mean that there should exist a functor $I \colon \mathcal{A}...

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### Abelian category equivalent to a non-abelian category [closed]

I was told that if we have an equivalence of categories $F : \mathcal{A} \rightarrow \mathcal{B}$ with $\mathcal{A}$ abelian, then it is not necessarily true that $\mathcal{B}$ is also abelian.
I ...

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208 views

### Coreflective subcategories in Grothendieck/locally presentable categories

This question is a reference request for the following result or two results, which I believe are rather easy to prove.
Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\...

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**1**answer

2k views

### Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact,...

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### Characterization of geometric morphisms without referring explicitly to the left adjoint?

Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...

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235 views

### When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?

Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...

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### Category of representations of a tensor product algebra

Given two semisimple unital algebras $A$ and $B$, defined over $\mathbb{R}$ or $\mathbb{C}$, denote their categories of representations by $_A\mathcal{M}$ and $_B\mathcal{M}$ respectively. Can one ...

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413 views

### Subfunctor of internal Hom

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...

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361 views

### Concrete examples of Freyd-Mitchell embedding

I originally posted this on math.SE (https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding) but since it's been a few days I figured I would crosspost it here. ...

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323 views

### Adjoints for radical and socle functors

Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...

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### Grothendieck groupoid associated to bicategory

Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of ...

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### Motivation for definitions of donor and receptor in Salamander Lemma?

$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes:
$$\require{AMScd}\begin{...

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535 views

### Comultiplication on objects in an (abelian?) category

Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication.
After studying a little bit about co-...