# Questions tagged [abelian-categories]

The abelian-categories tag has no usage guidance.

230
questions

11
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0
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### Description of the canonical equivalence between Adelman's free abelian category and Freyd's free abelian category on an additive category?

Let $\mathcal A$ be an additive category. Then there is a free abelian category $F(\mathcal A)$ on $\mathcal A$. I'm aware of two constructions in the literature, and I'd like to relate them.
The ...

1
vote

0
answers

154
views

### Proving that the functor induced by some inclusion functor has a left adjoint

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}\subseteq \operatorname{Proj}(\mathcal{A})$ be a full additive subcategory of $\mathcal{A}$. We define the full subcategory $\mathcal{B(A)}$ of ...

10
votes

2
answers

542
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### Abelian categories satisfying AB5*

I could name on the spot a bunch of abelian categories satisfying AB5 but I cannot think of any that satisfies AB5*. That is, it should have all limits and the cofiltered limits are exact. Is there ...

3
votes

0
answers

75
views

### Does a functor preserving injectives also preserve K-injective complexes?

Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes?
For example, ...

4
votes

1
answer

367
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### Can higher G-theory of Noetherian schemes be computed by derived categories?

Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K_0({\mathcal A})\cong K_0(D^{b}(\mathcal A))$.
When we set $\mathcal A$ to ...

3
votes

1
answer

275
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### Is the subcategory of strict morphisms abelian?

Let $A$ be an additive category with kernels and cokernels. A morphism $f$ is called strict if the natural morphism from the coimage to the image is an isomorphism.
In Schneiders: Quasi-abelian ...

3
votes

1
answer

208
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### Can we define $\operatorname{Ext}$ groups in the category of short exact sequences?

Let $\mathcal A$ be an Abelian category. We can assume it has enough injectives. Let $\operatorname{SES}_{\mathcal A}$ denote the category, of which objects are short exact sequences in $\mathcal A$, ...

2
votes

1
answer

150
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### Non-cofiltered derived limits

As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...

12
votes

2
answers

1k
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### Abelian categories that are not monoidal

Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories admit (symmetric?) monoidal structure (of course, with the tensor product being additive, ...

2
votes

2
answers

505
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### Can a category be enriched over abelian groups in more than one way?

An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian ...

2
votes

1
answer

182
views

### Is every filtration on an abelian category strict?

It's well-known that the category of filtered objects in an abelian category is generally not an abelian category. One must generally add extra structure to ensure that all morphisms are strict for ...

8
votes

1
answer

370
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### Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?

Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on ...

10
votes

0
answers

842
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### Does every monoidal abelian category admit an exact, lax monoidal functor to abelian groups?

Let $\mathcal A$ be a (small) non-zero abelian category equipped with a monoidal structure $\otimes$ which is right-exact in each variable. (Maybe feel free to assume more if that makes things easier -...

1
vote

1
answer

205
views

### Sufficient condition for right exact functor to be a left adjoint

Disclaimer: I first tried to ask this question on stackexchange https://math.stackexchange.com/questions/4577320/sufficient-conditions-for-a-right-exact-functor-to-be-a-left-adjoint but I did not get ...

1
vote

1
answer

87
views

### Covering by generators

Let $\mathcal A$ be an abelian category which contains all colimits. Let $\mathcal P$ be a full subcategory of generating objects. You may assume them to be projective. Is it true that for any $X\in \...

1
vote

1
answer

187
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### On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal

$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak ...

6
votes

1
answer

273
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### Vanishing of higher limits

Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...

15
votes

3
answers

1k
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### How exotic can an infinite biproduct in an additive category be?

Let $C$ be a category with a zero object $0$, small products, and small coproducts. Let $(A_i)_{i \in I}$ be a (possibly infinite) list of objects. There is a canonical map $\amalg_{i \in I} A_i \to \...

4
votes

1
answer

343
views

### Why does the category of abelian groups satisfy the axiom AB6?

In his Tohoku article, Grothendieck says that the category $\mathbf{Ab}$ of abelian groups satisfies the axiom AB6, namely
"All small colimits exist in $\mathbf{Ab}$. Moreover for any index ...

1
vote

0
answers

141
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### "Interesting" examples of exact abelian subcategories of R-Mod

A somewhat vague question: for which rings there exist "interesting" exact abelian subcategories of $R-\operatorname{Mod}$ that are closed with respect to products? Actually, I would like ...

3
votes

1
answer

246
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### Derived Hom without injectives nor projectives

I am stuck with the following farce on derived Homs.
I have an abelian category $A$ and I showed that, given any two objects $X$ and $Y$ of $A$, the group of $1$fold extensions $\operatorname{Ext}^1_{...

7
votes

2
answers

309
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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

100
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### 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

115
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 ...

5
votes

0
answers

158
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 ...

8
votes

2
answers

327
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 $...

7
votes

1
answer

280
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 ...

3
votes

1
answer

187
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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

276
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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 ...

5
votes

0
answers

243
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 (...

9
votes

1
answer

541
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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
answers

104
views

### 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
votes

1
answer

244
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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 ...

5
votes

0
answers

164
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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
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115
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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
votes

0
answers

106
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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

214
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 $...

3
votes

1
answer

110
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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 ...

4
votes

0
answers

82
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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 ...

3
votes

0
answers

58
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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

147
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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

118
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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

176
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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

270
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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

95
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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

185
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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

240
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

0
answers

108
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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

301
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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

123
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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 ...