Questions tagged [abelian-categories]

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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 ...
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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 $\mathscr C$ ...
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  • 31
3 votes
1 answer
103 views

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 ...
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  • 105
5 votes
0 answers
108 views

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 ...
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9 votes
2 answers
267 views

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 $...
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  • 3,556
7 votes
1 answer
209 views

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 ...
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3 votes
1 answer
130 views

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, ...
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3 votes
1 answer
105 views

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 ...
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3 votes
0 answers
180 views

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 (...
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8 votes
1 answer
444 views

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 ...
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5 votes
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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 ...
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6 votes
1 answer
184 views

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 ...
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4 votes
0 answers
138 views

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 ...
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2 votes
0 answers
107 views

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 ...
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2 votes
0 answers
100 views

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 ...
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4 votes
1 answer
168 views

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 $...
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3 votes
1 answer
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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 ...
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3 votes
0 answers
61 views

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 ...
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2 votes
0 answers
39 views

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 ...
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5 votes
0 answers
136 views

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:\...
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2 votes
0 answers
116 views

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 ...
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3 votes
1 answer
130 views

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 ...
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4 votes
1 answer
184 views

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 \...
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2 votes
0 answers
60 views

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 \...
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2 votes
0 answers
178 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 ...
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8 votes
1 answer
213 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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4 votes
0 answers
103 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 ...
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8 votes
1 answer
241 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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2 votes
1 answer
119 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 ...
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7 votes
1 answer
726 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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2 votes
1 answer
207 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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4 votes
0 answers
159 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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7 votes
1 answer
515 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 $R$...
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19 votes
1 answer
408 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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0 votes
0 answers
63 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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2 votes
0 answers
118 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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3 votes
0 answers
109 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 $YD(H)$ ...
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6 votes
1 answer
828 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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4 votes
0 answers
165 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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5 votes
1 answer
208 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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9 votes
1 answer
355 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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3 votes
1 answer
249 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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2 votes
0 answers
108 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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7 votes
1 answer
214 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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8 votes
0 answers
174 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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  • 3,854
10 votes
1 answer
253 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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2 votes
0 answers
39 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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4 votes
0 answers
105 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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  • 4,562
7 votes
1 answer
429 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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1 vote
1 answer
90 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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