Questions tagged [abelian-categories]
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209
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7
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Deformation of (locally) ringed spaces and of their abelian categories of modules
I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example ...
0
votes
0
answers
92
views
Do systems of objects over a Grothendieck category form a Grothendieck category?
Let $\mathscr C$ be a Grothendieck category and let $I$ be a small category (not a preadditive category, just a small category). Is the category $\mathscr C^I$ of systems of objects in $\mathscr C$ ...
3
votes
1
answer
103
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Does left-exactness imply semi-additivity?
Let $\mathcal C$ and $\mathcal D$ be pre-additive (enriched over the abelian groups) categories and $F : \mathcal C \Rightarrow \mathcal D$ a functor which is left-exact in the sense that it preserves ...
5
votes
0
answers
108
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Is there something similar to Lawvere-Tierney topologies for Abelian categories?
Lawvere-Tierney topologies generalize the notion of local operators on a Topos from Sheaf toposes over a Grothendieck site to arbitrary Toposes.
However, while the special case of Sheaves of sets or ...
9
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2
answers
267
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Pullback and pseudoelements
Let $\mathcal{A}$ be an abelian category, and let $X$ an object of $\mathcal{A}$. Recall that a pseudoelement of $X$ is an equivalence class of arrows $X_1 \to X$, where $x_1 \colon X_1 \to X$ and $...
7
votes
1
answer
209
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Does the category of commutative and cocommutative Hopf algebras have enough injectives?
It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
3
votes
1
answer
130
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Group action on fibre functor
(I asked this question on mathstack here: https://math.stackexchange.com/questions/4413271/group-action-on-fibre-functor. After getting no response and being suggested in the comments to post it here, ...
3
votes
1
answer
105
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Derived functors of inverse limit in abelian categories?
I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$.
I suppose that $\mathscr C$ has direct sums. Given that my ...
3
votes
0
answers
180
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A 2-category of abelian categories?
Is there a reasonably well-behaved $2$-category of abelian categories, and if so, what are its objects (perhaps one needs to restrict one's attention to finite abelian categories ?), $1$-morphisms (...
8
votes
1
answer
444
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What are abelian categories enriched over themselves?
As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...
5
votes
0
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97
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Indecomposable objects in iterated functor categories
Let $\mathcal{O}$ be a DVR with a uniformizer $\pi$ and a finite quotient field $\mathbb{F}_q=\mathcal{O}/\pi$. Write $\mathcal{O}_r = \mathcal{O}/(\pi^r)$. Fix now an integer $r\geq 2$. We define ...
6
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1
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184
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Is any abelian category a subcategory of $\mathrm{Ab}^I$?
Motivation: define a concrete Abelian category as a category with a univalent and injective functor in $\mathrm{Ab}^I$ (such that all homological concepts in it coincide with simple set-theoretic ...
4
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0
answers
138
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On the not so clear relationship between torsion theories and localization for a newcomer
Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...
2
votes
0
answers
107
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When semi-simple subcategories "extend" to hearts of t-structures?
Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct ...
2
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0
answers
100
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Is this concept of a left-abelian category studied?
A category is abelian if it is preadditive and
it has a zero object,
it has all binary biproducts,
it has all kernels and cokernels, and
all monomorphisms and epimorphisms are normal.
Now we ...
4
votes
1
answer
168
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When is a thick subcategory the preimage of a weak Serre class under a homological functor?
Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $...
3
votes
1
answer
91
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A non-projective rigid object in an abelian monoidal category
What is an example of a rigid object $A$ in an abelian monoidal category $\mathcal{M}$ that is not projective as an object in $\mathcal{M}$? (Since $\mathcal{M}$ is abelian projective just means that ...
3
votes
0
answers
61
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Derived category supported in a Serre subcategory of a locally noetherian category
This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
2
votes
0
answers
39
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A question about simple and finitely-generated objects in Grothendieck categories
Let $\mathcal{E}$ be a Grothendieck category and consider the following conditions:
(LFG) $\mathcal{E}$ is locally finitely generated (that is, the finitely generated objects of $\mathcal{E}$ generate ...
5
votes
0
answers
136
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Extension groups in quotient categories
Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
2
votes
0
answers
116
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A finiteness condition for Grothendieck categories
Let $\mathcal{E}$ be a locally finitely generated Grothendieck category. Let us say that an object $X$ of $\mathcal{E}$ is finitely cogenerated if every set of subojects of $X$ whose intersection is ...
3
votes
1
answer
130
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Category of modules over internal monoid is abelian
I have asked the following question on MSE a few days ago, but without any success.
I am interested in proving the following statement:
Let $\mathcal{A}$ be a tensor category. Then the category of ...
4
votes
1
answer
184
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In a category with a projective generator, do morphisms from the generator determine the object?
I have a cocomplete abelian category $\mathcal C$ and two objects $X$, $Y$ in $\mathcal C$.
Further, $\mathcal C$ has a projective generator $P$. I have an isomorphism
$$ \mathcal C(P,X) \cong \...
2
votes
0
answers
60
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Extensions in a full subcategory
Let $\mathcal{C}$ be an abelian category (feel free to put more adjectives here) and $\mathcal{D}$ a full abelian subcategory closed under kernels and cokernels.
Then by definition for $A,B\in \...
2
votes
0
answers
178
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Small abelian categories and module categories - preservation of injective and projective objects
A soft question on small abelian categories:
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper
Wikipedia: "The article "Sur quelques points d'algèbre homologique" by ...
8
votes
1
answer
213
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Bounds on homological dimension of functor categories
Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying ...
4
votes
0
answers
103
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How can one characterize categories of exact functors?
Does there exist any intrinsic characterization of additive categories equivalent to $\operatorname{Ex}(A,Ab)$, that is, of exact functors from a small abelian category $A$ into abelian groups? Any ...
8
votes
1
answer
241
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Filling square to push-out in abelian category
Let $\mathcal{C}$ be an abelian category. In $\mathcal{C}$ we consider the diagram
\begin{array}{ccc}
A&&\\\
\downarrow&&\\\
C&\rightarrow&D
\end{array}
with arrows being ...
2
votes
1
answer
119
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Methods for finding complex for subobjects of homology
Let $\mathcal{C}$ be an abelian category and
$$ 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 ...
7
votes
1
answer
726
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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 ...
2
votes
1
answer
207
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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 $...
4
votes
0
answers
159
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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
1
answer
515
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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
1
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408
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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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0
answers
63
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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
0
answers
118
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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
0
answers
109
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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
1
answer
828
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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
0
answers
165
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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
1
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208
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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
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1
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355
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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
1
answer
249
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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
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0
answers
108
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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
1
answer
214
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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
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0
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174
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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
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1
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253
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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
0
answers
39
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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
0
answers
105
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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
1
answer
429
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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
1
answer
90
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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 $\...