Questions tagged [abelian-categories]
The abelian-categories tag has no usage guidance.
230
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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 ...
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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 ...
- 111
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2
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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 ...
- 365
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75
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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, ...
- 353
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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 ...
- 411
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1
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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 ...
- 1,794
3
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1
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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 $\mathcal A$, ...
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1
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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 ...
- 505
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2
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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, ...
- 914
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2
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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 ...
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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 ...
- 14.7k
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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 ...
- 5,647
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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 -...
- 56.7k
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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 ...
- 197
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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 \...
- 1,794
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1
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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 ...
- 154
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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)...
- 2,896
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3
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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 \...
- 56.7k
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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 ...
- 43
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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 ...
- 16.3k
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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_{...
- 1,419
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2
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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 ...
- 1,377
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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 $\mathscr C$ ...
- 31
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115
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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 ...
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158
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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 ...
- 1,783
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2
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327
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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 $...
- 3,654
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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 ...
- 3,001
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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
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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 ...
- 31
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243
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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 (...
- 1,402
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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 ...
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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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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 ...
- 5,647
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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 ...
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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 ...
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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 ...
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214
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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 $...
- 56.7k
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1
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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 ...
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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 ...
- 41
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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 ...
- 710
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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:\...
- 1,419
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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 ...
- 710
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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 ...
- 275
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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 \...
- 131
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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 \...
- 181
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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 ...
user122276
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1
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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 ...
- 2,138
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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 ...
- 16.3k
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1
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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 ...
- 463
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1
answer
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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 ...
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