# Questions tagged [abelian-categories]

The abelian-categories tag has no usage guidance.

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### Infinite Krull-Schmidt categories?

In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...

**4**

votes

**0**answers

64 views

### On existence of finitely generated projective generator with commutative endomorphism ring in ${}_R Mod$

Let $R$ be a ring with unity (not necessarily commutative). Let ${}_R Mod$ be the category of left $R$-modules. If ${}_R Mod$ has a projective generator with commutative endomorphism ring , then does $...

**7**

votes

**1**answer

269 views

### Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...

**3**

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**0**answers

58 views

### Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...

**4**

votes

**1**answer

279 views

### Kernels and cokernels of multicomplex homomorphisms

Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...

**7**

votes

**0**answers

60 views

### Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...

**4**

votes

**1**answer

134 views

### DGA for a general abelian category

Differential graded algebras dga-s are fundamental objects of study in homological algebra and category theory. On the nlab webpage, they are defined as follows:
a dga is a monoid in the symmetric ...

**3**

votes

**1**answer

112 views

### Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...

**2**

votes

**1**answer

112 views

### Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...

**11**

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**0**answers

181 views

### Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?

Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...

**2**

votes

**1**answer

75 views

### An immersion of $\text{Sub}(X) \to \text{EqRel}(X)$ in a Malcev category

In an abelian category, each subobject $A \stackrel{f}{\to} X$ individuate an equivalence relation $R(f) \to X^2$ which is given by the equalizer of $$X^2 \rightrightarrows X \to \text{Coker}(f). $$
...

**0**

votes

**1**answer

85 views

### Misunderstanding of the proof of the Embedding Theorem in Borceux

In the proof of the full exact embedding theorem of the "Handbook of Categorical Algebra 2" of F. Borceux at the step 2, there is something I do not understand. It is at page 82 just after the ...

**14**

votes

**1**answer

676 views

### Is the category of left exact functors abelian?

Recently I asked on Math Stack Exchange here, if the category $\mathbf{Lex(\mathcal{A,B})}$ of left exact functors between to abelian categories $\mathcal{A,B}$ is abelian?
In his thesis Des ...

**3**

votes

**1**answer

90 views

### Brauer group classifying some splitting categories

Notation: $k$ - field. "$k$-category" = $k$-linear abelian category. $Vect_k$ - the $k$-category of $k$-vector spaces. For a field extension $K/k$ and a $k$-category $\mathcal{A}$, denote by $\mathcal{...

**8**

votes

**1**answer

436 views

### Example of an abelian category with enough projectives and injectives which are not dual

For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of ...

**29**

votes

**1**answer

1k views

### What was the error in the proof of Roos' theorem?

Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...

**4**

votes

**0**answers

147 views

### On representable abelian sheaves vs abelian sheaves

Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable
(either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.
Suppose there is ...

**10**

votes

**2**answers

945 views

### Is every “nice” abelian category with enough projectives an additive presheaf category?

A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...

**24**

votes

**1**answer

1k views

### Locally presentable abelian categories with enough injective objects

I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant.
Does there exist a locally presentable abelian ...

**1**

vote

**1**answer

155 views

### Abelian category which is nice and its dual is also

Is there a nontrivial abelian category $C$ such that both $C$ and $C^{op}$ satisfy AB3-AB5? ("nontrivial" means that there are nonzero objects in $C$)

**38**

votes

**4**answers

4k views

### What's there to do in category theory?

Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I ...

**4**

votes

**0**answers

107 views

### Category of (co)commutative Hopf monoids in an exact category

I'm transferring this question over from SE, since it didn't get much attention over there.
Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...

**8**

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289 views

### Abelian categories have become the language of homological algebra. Why haven't Zariski categories become the language of commutative algebra?

I'm not seeing much mention of Zariski categories in the literature. There is no article on Zariski categories in nLab, which would seem to be an obvious place to have such an article. What has ...

**3**

votes

**0**answers

129 views

### About functor categories and enough injectives [duplicate]

Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove ...

**7**

votes

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128 views

### Quotifiable Subcategories

I asked this on math.stackexchange but haven't got a response. If this question is unsuitable for this website please let me know.
In Popescu's textbook Abelian Categories with Applications to Rings ...

**2**

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**0**answers

128 views

### Are finitely generated objects in a locally Noetherian category also finitely presented?

Let $\mathcal C$ be a locally Noetherian category. Then, we know that every finitely generated object is Noetherian.
I cannot seem to be able to prove that Noetherian objects are finitely presented. ...

**8**

votes

**0**answers

243 views

### Is every $R^iF(M)$ isomorphic to some $F(N)$?

Let $A$ and $B$ be abelian categories. Assume that $A$ has enough injectives. Is there a "useful" (take that to mean what you will) condition on $A$ and $B$ such that the following is true?
For all ...

**3**

votes

**1**answer

207 views

### A simple colimit in the derived category?

I have recently come across the following question :
Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma_{\leq n}}$ the stupid truncations functors (i.e. ...

**6**

votes

**2**answers

453 views

### Direct sum of injective modules is injective

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...

**3**

votes

**1**answer

149 views

### Exactness of $j_!$ in abelian category recollement

Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...

**9**

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**0**answers

344 views

### How to Compute Ext-Groups for Categories without Enough Injectives/Projectives?

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...

**4**

votes

**0**answers

169 views

### Checking a monad is idempotent

I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...

**2**

votes

**0**answers

163 views

### Why holim and not Rlim?

Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\...

**2**

votes

**1**answer

360 views

### Let $F:\mathscr{A}\to\mathscr{B}$ be an equivalence of Abelian categories. Must $F$ be additive?

What if we simply require $\mathscr{A}$ to be pre-additive, or additive? I have seen it stated without proof that if $\mathscr{A}$ and $\mathscr{B}$ are abelian categories, then any equivalence $F:\...

**3**

votes

**1**answer

481 views

### On an exercise from Weibel's book on homological algebra

I was revising some old postgraduate notes of mine in homological algebra (written during a postgrad course on the topic, I had taken more than ten ;) years ago) and I came accross the following ...

**1**

vote

**1**answer

150 views

### Is the minimal Serre subcategory containing a set also a set?

Let $\mathscr{A}$ be an abelian category, and $S\subset\mathscr{A}$ some set of objects in $\mathscr{A}$ (i.e. $S$ is not a proper class), so we can fairly easily define the category $\mathscr{C}_S$ ...

**4**

votes

**1**answer

287 views

### Colimits in colimits of categories

I was considering the following situation. I have a directed/filtering system of categories $C_i$. I understand how take its direct limit (aka colimit) $C$ in the category of categories. My question ...

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**0**answers

294 views

### Is every Artinian abelian category necessarily Noetherian?

Let $\mathcal{A}$ be a small abelian category (additive category with AB1) and AB2)). We say $\mathcal{A}$ is artinian if for every $A\in\mathcal{A}$, every descending chain of subobjects of $A$ ...

**28**

votes

**3**answers

4k views

### What is a triangle?

So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...

**3**

votes

**1**answer

267 views

### Does the existence of an injective cogenerator “help” in finding generators of an abelian category?

I have an abelian category $A$ that is AB4, AB3* and has an injective cogenerator. Do these conditions "help" in checking whether a given family $a_i$ of (compact) objects of $A$ is generating in it? ...

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1k views

### Can $\mathcal O_X$ be recognized abstract-nonsensically?

This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.
In the ...

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**0**answers

129 views

### Names for $\operatorname{Coker}\operatorname{ker}\cong \operatorname{Ker}\operatorname{coker}$ and $\operatorname{Coim}\cong \operatorname{Im}$?

Let $\operatorname{Coim}f$ be the cokernel of the kernel pair of an arrow. Let $\operatorname{Im}f$ be the kernel of the cokernel pair.
An interesting property of a category is the isomorphy of the ...

**9**

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**1**answer

461 views

### Objects of which Grothendieck abelian categories have elements?

The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those ...

**4**

votes

**2**answers

454 views

### On various relations between “additional axioms” for AB4 and Grothendieck abelian categories

Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list ...

**4**

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**0**answers

148 views

### Relationship between pointed protomodularity and $\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$

In hopes of understanding algebra better, I've been reading here and there about protomodular categories and the like. Among other things, the theory surrounding these (and some other) categories ...

**19**

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269 views

### Homotopic version of Freyd's AT category observations

Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...

**2**

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**0**answers

200 views

### On the category of $D$-modules

Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$.
1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...

**2**

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**1**answer

175 views

### Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...

**4**

votes

**1**answer

129 views

### Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...

**3**

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**1**answer

332 views

### Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\...