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.

learn more… | top users | synonyms

1
vote
1answer
172 views

Characterization of injective objects in abelian categories

In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
3
votes
1answer
208 views

Is every right adjoint of a surjective functor fully faithful?

Let $F\colon C\rightarrow D$ a functor, which is surjective and $G$ a right adjoint of $F$. Is $G$ always fully faithful? What if $C,D$ are abelian and $F$ is additive? I know that it is always ...
5
votes
1answer
340 views

Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”

Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says: Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and ...
3
votes
1answer
65 views

Showing the intrinsic addition of an Abelian Category is associative

Let $\mathcal A$ be an abelian category. Let $(P\oplus Q,\eta_1,\eta_2)$ be the coproduct of $P$ and $Q$, then if $\delta_1=[id_P,0]$ and $\delta_2=[0,id_Q]$, then $(P\oplus Q,\delta_1,\delta_2)$ is ...
2
votes
1answer
82 views

Proof of exactness at the first two non-zero objects in the ker-coker sequence (snake lemma).

I am reading MacLane's chapter on Abelian Categories and I am proving the fact, needed for the snake lemma, that the sequence $0\to \text{Ke}f\to \text{Ke}g\to\text{Ke}h$ is exact at $\text{Ke}f$ and ...
1
vote
2answers
48 views

Does a linear quotient map have sections

Suppose $V$ is a vector space with vector subspace $N$. Then there is a natural projection $$ \pi_N: V \to V/N $$ from the vector space $V$ to the quotient space $VN$ of $V$ modulo $N$. Does ...
4
votes
2answers
135 views

(AB1) failure in $\mathcal{K(A)}$, without triangulated categories

I am trying to prove that there exists an abelian category $\mathcal A$ such that its homotopy category $\mathcal{K(A)}$ is (additive but) not abelian, without passing through triangulated categories ...
3
votes
1answer
77 views

Is every additive monofunctor between abelian categories left exact?

Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
3
votes
2answers
126 views

Simple question on exact sequences.

I am just learning about abelian categories and I would like to hear some advice on how to think about the concepts of kernel, cokernel, image etc correctly; I am a bit confused. I am trying to prove ...
5
votes
0answers
139 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 ...
1
vote
2answers
212 views

Questions about epimorphisms and projectives in functor categories

Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
2
votes
0answers
52 views

Definition of Exact functors [duplicate]

As we know by definition, a functor for example $T\colon R\textrm{-}\mathsf{Mod}\to\mathsf{Ab}$ (from the category of $R$-modules to the category of abelian groups) is "exact" precisely when for any ...
0
votes
3answers
160 views

Abelian categories with direct sums

Does any abelian category admits direct sums? If not, categories admiting direct sums have a special name? I'm asking this since I am writing a proof that requires direct sums but I only know that ...
1
vote
1answer
230 views

What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
5
votes
2answers
228 views

Applications of Mitchell's embedding theorem

I don't understand what is the advantage of viewing a particular category as a category of modules over some ring. Can anybody tell me some application of Mitchell's embedding theorem so that I can ...
1
vote
1answer
414 views

The definition of the quotient category in abelian category.

I want to understand the definition of morphisms in this category. My question is how can I construct directed sets and direct systems, and therefore understanding the colimite. Definition: Given a ...
5
votes
1answer
205 views

Additive category and zero map

Let $A$ be an additive category. Namely $A$ has a zero object, $A$ has finite products and coproducts, and Every Hom-set is an Abelian group such that composition of morphisms is bilinear. ...
3
votes
0answers
92 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 ...
45
votes
1answer
2k views

Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
0
votes
0answers
161 views

Long Exact Sequence on Homology in an Abelian Category

Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
4
votes
1answer
108 views

Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
1
vote
1answer
110 views

Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction

Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, ...
2
votes
2answers
112 views

MacLane's Abelian Categories Chapter

I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of ...
2
votes
1answer
290 views

split exact complexes and biproducts

I have an split exact complex in an abelian category, that is, a chain complex which is exact and maps $s_n : C_n \to C_{n+1}$ st. $dsd = d$. I would like to prove that this implies that $C_n \cong ...
2
votes
1answer
95 views

Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?

Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$. Why: (1) ...
5
votes
2answers
352 views

intersections in abelian category

Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow ...
1
vote
0answers
165 views

Is $\mathbb{AB}$ an additive (or even abelian) category?

Notation. Let $U_0 \in U_1 \in U_2$ be Grothendieck universes, each containing $\mathbb N$. Let $\mathbf{Cat}_{U_0}$ be the ($U_1$-small) 2-category of all $U_0$-small categories, ...
0
votes
1answer
181 views

Difficult category theory: kernels

In a category with a zero object ($0$ = initial and terminal) and zero morphisms are unique $A \to 0 \to B$ for every $A,B$, define the kernel of a map as the equalizer of $(f,0)$ and cokernel dually. ...
2
votes
1answer
134 views

If an abelian category has a generator then it is well-powered

The title of my question is Proposition 3.35 in Freyd's Abelian categories. The proof says If $G$ is a generator and $A$ is any object, then a subobject $A' \longrightarrow A$ is distinguished by ...
4
votes
1answer
224 views

Equivalent characterizations of faithfully exact functors of abelian categories

Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question. Let $X \rightarrow Y \rightarrow Z$ be a ...
0
votes
0answers
44 views

What do we call a functor which is exact and reflects exact sequences in abelian categories? [duplicate]

Possible Duplicate: what is a faithfully exact functor? Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We say $F$ reflects exact sequences if $F$ ...
3
votes
1answer
145 views

Free abelian groups and abelian categories

Why is the category of free abelian groups not an abelian category?
10
votes
2answers
845 views

If a functor between categories of modules preserves injectivity and surjectivity, must it be exact?

Let $A$ and $B$ be commutative rings. Let $F$ be a functor from the category of $A$ modules to the category of $B$ modules. Suppose that $F$ preserves injectivity and surjectivity: whenever $f : ...
2
votes
0answers
131 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
1
vote
1answer
244 views

Exactness of Colimits

Let $\mathcal A$ be a cocomplete abelian category, let $X$ be an object of $\mathcal A$ and let $I$ be a set. Let $\{ X_i \xrightarrow{f_i} X\}_{i \in I}$ be a set of subobjects. This means we get an ...
1
vote
1answer
68 views

Smallest subobject in an abelian category containing a set of objects

Let $\mathcal A$ be an abelian category. Let $A$ be an object in $\mathcal A$ and let $(A_i)_{i\in I}$ be a set of subobjects of $A$. Then there is a subobject $\sum_{i \in I} A_i$ of $A$ which has ...
2
votes
1answer
260 views

Set of generators in an abelian category - two definitions

Let $\mathcal C$ be a category. We say that $\mathcal C$ has a set of generators $\{ G_i\}_{i \in I}$ if whenever we take two distinct morphisms $f, g \colon A \to B$ in $\mathcal C$ there exists some ...
4
votes
1answer
204 views

Category of vector bundles over a space is additive?

Consider the category of vector bundles over a fixed base space. Then this category is not abelian, since the kernel of a morphism of bundles is in general not a vector bundle. But is it additive? ...
3
votes
2answers
245 views

An exact sequence of homology in abelian categories

$\renewcommand{\im}{\mathop{\rm im}}\DeclareMathOperator{\coker}{coker}$Let $A\xrightarrow{f}B\xrightarrow{g}C$ be a complex in an abelian category, I.e. $gf=0$. Let $H:=\coker(\im(f)\to \ker(g)).$ ...
0
votes
0answers
73 views

Image in abelian categories [duplicate]

$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$For a morphism $ f: A\to B$ in an abelian category, we let $\im f:=\ker(\coker f)$. Then the morphism $A\to \im f$ is an epimorphism and ...
2
votes
2answers
129 views

What's the definition of a free module in an abelian category?

I was trying to prove that free modules were projective in the language of abelian categories, but did not succeed. I was missing a good description of what a free module is. So my question is ...
12
votes
1answer
522 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
9
votes
1answer
522 views

The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
20
votes
1answer
1k views

When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then ...
33
votes
3answers
1k views

Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
3
votes
1answer
575 views

Definition of a Functor of Abelian Categories

What is the precise definition of a functor of abelian categeries. I've looked on the internet but can't find one. From the Wikipedia definition of an abelian category, I'm guessing that, for two ...
3
votes
0answers
87 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 ...
1
vote
1answer
128 views

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
1
vote
0answers
78 views

invert Grothendieck spectral sequence

I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves) $F: A\rightarrow B$ $G: B \rightarrow C $ $H: A ...
35
votes
2answers
1k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...