Questions tagged [abelian-categories]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
0 answers
124 views

Is taking Freyd envelopes adjoint to taking stable module categories?

Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
Tim Campion's user avatar
  • 60.5k
1 vote
1 answer
110 views

Reference request regarding faithfully exact functors between abelian categories

I am looking for a reference for the following result (or any subresult) in any book or notes: Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
122 views

Dual objects in an abelian monoidal category

Let $(\mathcal{M},\otimes)$ be a monoidal category that is also abelian, and assume that $\otimes$ is bilinear with respect to the biproduct. Let $X$ and $Y$ be two objects that admit (left) duals ${}^...
Yilmaz Caddesi's user avatar
5 votes
0 answers
173 views

Is there a way to “derive” a (non-exact) functor which preserves images?

Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure ...
Tim Campion's user avatar
  • 60.5k
3 votes
1 answer
139 views

Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
Ji Woong Park's user avatar
7 votes
0 answers
170 views

Why are Gabriel categories closed under localization?

Let $\mathcal K$ be a Grothendieck category. Recall the Gabriel filtration $0 \subseteq \mathcal K_1 \subseteq \cdots \mathcal K$ of localizing subcategories, where $\mathcal K_{\alpha+1}$ is ...
Tim Campion's user avatar
  • 60.5k
10 votes
2 answers
872 views

Example of a Grothendieck category which is not Gabriel?

Following Gabriel, for $\mathcal C$ a Grothendieck category, set $\mathcal T(\mathcal C)$ to be the localizing subcategory generated by the objects of finite length, and $\mathcal C' = \mathcal C / \...
Tim Campion's user avatar
  • 60.5k
2 votes
0 answers
115 views

Left duals and right duals are also isomorphic in a semisimple category

In the n-Lab page https://ncatlab.org/nlab/show/rigid+monoidal+category it is written that Left duals and right duals are also isomorphic in a semisimple category. For a left dual semisimplicity ...
Yilmaz Caddesi's user avatar
5 votes
1 answer
218 views

Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
Snake Eyes's user avatar
2 votes
1 answer
72 views

Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?

The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
Tim Campion's user avatar
  • 60.5k
2 votes
1 answer
148 views

How does the behaviour of a hyperderived functor of many variables change if you use $\prod$-totalisation instead of $\oplus$-totalisation?

$\newcommand{\tot}{\operatorname{Tot}}\newcommand{\A}{\mathscr{A}}\newcommand{\L}{\mathbb{L}}\newcommand{\R}{\mathbb{R}}$Say $T$ is is a functor $\A_1\times\A_2\times\cdots\times\A_n\to\A$ of Abelian ...
FShrike's user avatar
  • 487
7 votes
0 answers
246 views

What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...
David Corwin's user avatar
  • 15.1k
6 votes
1 answer
281 views

What conditions on an Abelian category allow members of a direct sum to be determined entirely by their components?

EDIT: In comments, with thanks to Maxime Ramzi, this question has a good answer in that what I want to be true is true when $\mathscr{A}$ satisfies axiom $\mathsf{AB}5$, that $\mathscr{A}$ is ...
FShrike's user avatar
  • 487
6 votes
1 answer
523 views

In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?

Question: Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact ...
Tim Campion's user avatar
  • 60.5k
2 votes
0 answers
64 views

What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category). Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
Tim Campion's user avatar
  • 60.5k
3 votes
1 answer
257 views

A question about rigid objects in monoidal categories

Let $(\mathcal{A},\otimes,1_{\mathcal{A}})$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by $$ \Big\{M^*,~~~ \epsilon_l:M^* \otimes M \to ...
Yilmaz Caddesi's user avatar
2 votes
1 answer
207 views

Projective objects in chain complexes of an abelian category: Further question

Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes I am wondering why a level-wise projective chain complex $P$ which is split ...
locally trivial's user avatar
7 votes
0 answers
140 views

An abelian category with a full embedding from topological abelian groups

I know this is a very vague question, but I can't think of a better question to ask. Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \...
Charles Wang's user avatar
11 votes
0 answers
104 views

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 ...
Tim Campion's user avatar
  • 60.5k
2 votes
0 answers
183 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 ...
Juan C. Cala's user avatar
9 votes
2 answers
590 views

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 ...
user141099's user avatar
3 votes
0 answers
112 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, ...
Doug Liu's user avatar
  • 433
4 votes
1 answer
377 views

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 ...
Boris's user avatar
  • 501
3 votes
1 answer
300 views

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 ...
user avatar
3 votes
1 answer
222 views

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 $\mathcal A$, ...
Display Name's user avatar
2 votes
1 answer
160 views

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 ...
Matteo Casarosa's user avatar
12 votes
2 answers
1k views

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, ...
xuq01's user avatar
  • 1,032
2 votes
2 answers
518 views

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 ...
Didier de Montblazon's user avatar
2 votes
1 answer
201 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 ...
David Corwin's user avatar
  • 15.1k
8 votes
1 answer
386 views

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 ...
Arshak Aivazian's user avatar
9 votes
0 answers
880 views

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 -...
Tim Campion's user avatar
  • 60.5k
1 vote
1 answer
281 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 ...
Adelhart's user avatar
  • 227
1 vote
1 answer
92 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 \...
user avatar
1 vote
1 answer
210 views

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 ...
uno's user avatar
  • 280
6 votes
1 answer
329 views

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)...
AlexE's user avatar
  • 2,926
15 votes
3 answers
1k views

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 \...
Tim Campion's user avatar
  • 60.5k
4 votes
1 answer
480 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 ...
rtwo's user avatar
  • 43
1 vote
0 answers
146 views

"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 ...
Mikhail Bondarko's user avatar
3 votes
1 answer
293 views

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_{...
Stabilo's user avatar
  • 1,469
7 votes
2 answers
346 views

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 ...
AT0's user avatar
  • 1,437
0 votes
0 answers
102 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 $\mathscr C$ ...
FDR's user avatar
  • 41
3 votes
1 answer
123 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 ...
javra's user avatar
  • 105
5 votes
0 answers
186 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 ...
saolof's user avatar
  • 1,823
8 votes
2 answers
350 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 $...
Ricky's user avatar
  • 3,674
7 votes
1 answer
302 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 ...
Avi Steiner's user avatar
  • 3,031
3 votes
1 answer
203 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, ...
angry_math_person's user avatar
4 votes
1 answer
379 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 ...
FDR's user avatar
  • 41
5 votes
0 answers
283 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 (...
Dat Minh Ha's user avatar
  • 1,472
9 votes
1 answer
585 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 ...
Arshak Aivazian's user avatar
5 votes
0 answers
106 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 ...
Ehud Meir's user avatar
  • 4,969

1
2 3 4 5