Questions tagged [abelian-categories]

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2
votes
1answer
77 views

When uniquely divisible objects can be embedded into ind-torsion ones?

Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid_M$ is invertible. We will say that $M'$ is ind-...
5
votes
1answer
210 views

Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says: 'Conversely, it is well known (and easy to show) that any exact faithful functor ...
4
votes
1answer
127 views

Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian? [stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)] In ...
3
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0answers
50 views

Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
4
votes
1answer
124 views

Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
10
votes
1answer
194 views

Freyd-Mitchell for $k$-linear categories

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of ...
1
vote
0answers
47 views

Condition for an additive functor to be an equivalence

Consider an additive functor $F : \mathcal{A} \longrightarrow \mathcal{B}$ between abelian categories and suppose that $F$ is a dense functor, that is, for every object $B$ in $\mathcal{B}$ there is ...
4
votes
2answers
229 views

Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
3
votes
0answers
80 views

Baer sum and endomorphisms

I work in an Abelian category. If I take the Baer sum $M' + M''$ of two extensions $M'$ and $M''$ of $ M_2$ by $M_1$, i.e., $$ 0 \to M_1 \to M' \to M_2 \to 0$$ is exact, and the same for $M''$, then ...
14
votes
1answer
489 views

Abelian category with enough injectives but not functorially

Let $\mathcal{A}$ be an Abelian category with enough injectives. Is it always possible to make the injective embedding functorial? By this I mean that there should exist a functor $I \colon \mathcal{A}...
6
votes
3answers
1k views

Abelian category equivalent to a non-abelian category [closed]

I was told that if we have an equivalence of categories $F : \mathcal{A} \rightarrow \mathcal{B}$ with $\mathcal{A}$ abelian, then it is not necessarily true that $\mathcal{B}$ is also abelian. I ...
3
votes
1answer
164 views

Coreflective subcategories in Grothendieck/locally presentable categories

This question is a reference request for the following result or two results, which I believe are rather easy to prove. Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\...
12
votes
1answer
732 views

Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ? Does it, in fact,...
10
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0answers
96 views

Characterization of geometric morphisms without referring explicitly to the left adjoint?

Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
5
votes
1answer
223 views

When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?

Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
4
votes
1answer
156 views

Category of representations of a tensor product algebra

Given two semisimple unital algebras $A$ and $B$, defined over $\mathbb{R}$ or $\mathbb{C}$, denote their categories of representations by $_A\mathcal{M}$ and $_B\mathcal{M}$ respectively. Can one ...
4
votes
2answers
386 views

Subfunctor of internal Hom

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...
5
votes
2answers
288 views

Concrete examples of Freyd-Mitchell embedding

I originally posted this on math.SE (https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding) but since it's been a few days I figured I would crosspost it here. ...
4
votes
2answers
264 views

Adjoints for radical and socle functors

Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
2
votes
0answers
113 views

Grothendieck groupoid associated to bicategory

Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of ...
2
votes
0answers
72 views

Motivation for definitions of donor and receptor in Salamander Lemma?

$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes: $$\require{AMScd}\begin{...
5
votes
1answer
472 views

Comultiplication on objects in an (abelian?) category

Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-...
1
vote
0answers
46 views

Characterization of weakly convergence of spectral sequences

Let $C$ be a chain complex (in any abelian category) and let $\{F_p\}$ be a decreasing filtration of $C$. It induces a filtration on the homologies of $C$, given by $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
3
votes
0answers
55 views

Categorical construction of comodule category of FRT algebra

Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be ...
5
votes
0answers
118 views

Relation between extensions and filtrations

We work in an Abelian category. Consider Yoneda extensions, i.e., the Abelian groups Ext$^n(C,A)$ consisting (for $n \ge 1$) of equivalence classes of exact sequences starting at $A$ and ending at $C$ ...
1
vote
0answers
118 views

Injective envelope in the category of left exact functors

Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
2
votes
2answers
293 views

Motivation/intuition behind the definition of delta-functors and related concepts

I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter. Why are $\delta$-functors ...
5
votes
1answer
166 views

Does every functor between Grothendieck categories have adjoints?

Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits. Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories. Does $F$ have a right adjoint? ...
4
votes
2answers
240 views

Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?

Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...
3
votes
1answer
149 views

Ext-vanishing in abelian categories

Given an abelian category $A$ with enough projectives and enough injectives such that projectives do not coincide with injectives. Can we have $Ext^i(I,P)=0$ for any $i>0$ and injective $I$ and ...
3
votes
0answers
77 views

A class of Grothendieck categories

Is there a name and some work on Grothendieck categories in which each non-zero object has a simple subquotient? Every locally finitely generated Grothendieck category has this property, but I guess ...
8
votes
0answers
195 views

Is every Grothendieck category with a generator a category of sheaves?

The Gabriel-Popescu theorem tells us that every Grothendieck category with a generator is a left exact localization of a module category. I'm interested in a slightly different way of "representing" ...
4
votes
1answer
410 views

Do coherent sheaves on rigid analytic spaces form an abelian category?

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...
3
votes
0answers
63 views

Seeking for a counter-example for regularity properties of abelian categories

Can one construct a Grothendieck category which has enough projective objects, which is also locally finitely generated (or maybe even locally noetherian), but which does not have enough finitely ...
3
votes
0answers
154 views

Locally noetherian (AB4*) Grothendieck categories

Let A be a Grothendieck category satisfying (AB4*) (that is, with exact products). Assuming that, moreover, A is locally noetherian, must this category have enough small projective objects (so, to be ...
10
votes
3answers
756 views

Name for abelian category in which every short exact sequence splits

What is the name of the class of abelian categories defined by the following property: every short exact sequence splits?
6
votes
1answer
243 views

Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(A\otimes B) \longrightarrow N_\ast(A)\otimes N_\ast(B)$$ and ...
2
votes
0answers
111 views

Coend of full subcategory

$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over $...
7
votes
2answers
278 views

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 $r = s$, ...
5
votes
0answers
93 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
1answer
463 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 $\...
4
votes
0answers
72 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
1answer
301 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
0answers
64 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
1answer
171 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
1answer
176 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
1answer
149 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 (...
12
votes
0answers
444 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
1answer
95 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
1answer
93 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 ...