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Questions tagged [abelian-categories]

Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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870 views

Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
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642 views

Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...
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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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226 views

Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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Is there a finite abelian category?

Is there a non-discrete abelian category which has only finitely many objects? Just out of curiosity I am wondering if such an abelian category exists, while the usual examples of abelian categories ...
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459 views

What's there to do in category theory?

I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it. Now ...
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445 views

Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
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146 views

Derived functors of abelian categories via model categories.

I am trying to reproduce to me familiar methods of homological algebra using the language of model categories, but I run into a few small problems. Consider a left exact functor $F: \mathcal{A} \to \...
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Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
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445 views

Mistake in contradiction argument to show that sheafification commutes with cokernel

The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
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69 views

How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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Noetherian Objects in Quotient Category

I've recently been learning about the quotient of an abelian category by a Serre (thick) subcategory, specifically in the context of module categories. A thought occurred to me today about ...
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339 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 ...
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147 views

Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title $E_2^{p,q}=\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{...
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447 views

A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.1) Suppose $A$ is a (not necessarily bounded below) chain complex over an abelian category $\mathcal A$ where axiom (AB4) holds, ...
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172 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
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220 views

Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon \...
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375 views

A chain complex is exact iff it is split

Let $C.$ be a chain complex in an abelian category. Suppose that the identity $Id_{C.}$ is null homotopic. Then there are morphisms $s_n:C_n\to C_{n+1}$ such that for every $n$ the following identity ...
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92 views

Short proof or wrong proof?

I found in Borceux' Categorical Algebra the following proposition: in an abelian category, the following are equivalent: 1) $f:A\longrightarrow B$ is a mono 2) $\operatorname{Ker} f=0$ 3) for ...
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When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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46 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow F(M\...
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Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...
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618 views

Why homotopy category is not abelian?

Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are ...
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127 views

Exercise in Homological Algebra

I'm totally stuck with this problem that I found in an Algebra course. It is the following: Let $F:\mathcal{A} \to \mathcal{B}$ be a left exact functor between two abelian cathegories. Let $\...
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32 views

A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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43 views

Uniquenes of decompositions in abelian semisimple categories

Basing my intuition on the semisimple Lie algebra case, I have a question about semisimple abelian categories. Let $\mathcal{C}$ be such a category, and let $X$ be an object in $\mathcal{C}$. By ...
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131 views

extending functors

In "Functors on locally finitely presented additive categories", by H. Krause, one can read (in page 108): Proposition 2.3. There is, up to equivalence, a bijective correspondence between (skeletally ...
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66 views

Abelian categories linear over a field

A category $\mathcal{A}$ is abelian if it has the following properties: There is a zero object (both initial and terminal). Every pair of objects has a product and a coproduct (which will be ...
3
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0answers
68 views

What does an additive, mono-preserving functor have to do to become left exact?

Consider an additive functor $F \colon C → D$ of abelian categories preserving monos. As Martin has demonstrated here, $F$ may not be left exact. If $F$ even preserves kernels, $F$ is left exact. But ...
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59 views

Quotifiable Subcategory

In Popescu's textbook Abelian Categories with Applications to Rings and Modules, Theorem $3.3$ of section $4.1$ says: Let $\mathscr A$ be a dense subcategory of a locally small abelian category $\...
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189 views

Proving Kernels and Cokernels exist in Category of Short exact Sequences

Suppose $A$ is an abelian category. Then, we can define a category $Ses(A)$ of short exact sequences in $A$. I proved that the category is additive but I am not sure how to prove that this category ...
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135 views

What can we say about the category of spectral sequences?

For simplify, let's just consider the following category first: object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian ...
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137 views

Diagram chasing in Abelian categories?

In the nLab page, a technique so-called generalized elements is introduced, which is identical to that on MacLane's Categories for the Working Mathematician. We know that in this method, one can check ...
3
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52 views

Does Projectiveness always imply flatness?

I know that a project module is always flat, deduced form the properties and abundance of free modules. I'm trying to figure out how essential role the free modules play in this result. So I'd like to ...
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0answers
76 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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68 views

An abelian category such that all objects are injective

The problem is 'Let C be an abelian category such that all objects in C are injective. Prove that all abjects are projective.' If C has enough projectives, then the 'Ext' functor can be defined. Thus ...
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108 views

Show that homology is a functor in a pure categorical way.

Let $\mathscr{A}$ be an abelian category i want to show that $\mathcal{H^i}$ ( the i-th homology group) is a functor from the category of complexes of $\mathscr{A}$ to $\mathscr{A}$. I showed this for ...
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69 views

How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here $\underline{R-\mathrm{mod}}...
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172 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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92 views

Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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0answers
490 views

Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
3
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136 views

Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all $\overline{\mathbb{Q}_{\ell}}$-...
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0answers
107 views

A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal A$...
3
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0answers
103 views

Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
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31 views

How much can the Freyd-Mitchell embedding theorem be improved?

Can the embedding theorem be improved to preserve injective objects or even better be part of an adjoint pair? Or if that's not possible are there stronger conditions (AB5, enough injectives, etc) on ...
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31 views

Products, coproducts and morphisms

The universal properties of products and coproducts "amount" to the statements $$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$ ...
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81 views

Derived functors commute with filtered colimits?

I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer. ...
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0answers
51 views

Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
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0answers
63 views

What are some really weird abelian categories?

Once one studies algebra, one finds categories such as $R-\textbf{Mod}$, abelian groups, sheaves over abelian groups, $\mathcal R-\textbf{Mod}$ and the like. They are all abelian. On the other hand, ...
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72 views

constructing a right adjoint to i:eff C ---> mod C

In "Deriving Auslander's Formula (Theorem 2.2)", I'm trying to construct a right adjoint to inclusion functor $\mathsf{i:eff~C \to mod~C}$. I constructed it as follows: The functor $\mathsf{j:mod~C \...