Questions tagged [abelian-categories]
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185
questions
2
votes
0answers
133 views
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 ...
8
votes
1answer
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 ...
4
votes
0answers
86 views
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 ...
8
votes
1answer
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 ...
2
votes
1answer
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
votes
1answer
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 ...
3
votes
1answer
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 $...
3
votes
0answers
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 ...
7
votes
1answer
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
votes
1answer
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-...
0
votes
0answers
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 ...
2
votes
0answers
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}\...
3
votes
0answers
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
votes
1answer
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 ...
4
votes
0answers
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 ...
5
votes
1answer
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 $A^{op}$ ...
9
votes
1answer
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-...
3
votes
1answer
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 ...
2
votes
0answers
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 ...
7
votes
1answer
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 \...
8
votes
0answers
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 \...
10
votes
1answer
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 ...
2
votes
0answers
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 ...
4
votes
0answers
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 $\mathcal{C}$ ...
7
votes
1answer
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 ...
1
vote
1answer
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 $\...
11
votes
0answers
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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-...
7
votes
1answer
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 ...
4
votes
1answer
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 ...
3
votes
0answers
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 ...
4
votes
1answer
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 ...
9
votes
1answer
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
votes
1answer
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 ...
7
votes
2answers
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 ...
3
votes
0answers
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 ...
14
votes
1answer
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}...
6
votes
3answers
1k views
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 ...
4
votes
1answer
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\...
13
votes
1answer
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,...
12
votes
0answers
105 views
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 ...
5
votes
1answer
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{...
4
votes
1answer
170 views
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 ...
4
votes
2answers
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{...
5
votes
2answers
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. ...
5
votes
2answers
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 ...
2
votes
0answers
115 views
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 ...
2
votes
0answers
82 views
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{...
5
votes
1answer
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-...
