Questions tagged [abelian-categories]

Use this tag for questions about Abelian categories, which are categories that possess most of properties of categories of modules over a ring, and are easy to work with using techniques of homological algebra.

Definition

The definition of abelian category is defined so as to model all the nice properties that a category of modules over a ring has. Explicitly, a category $\mathcal{C}$ is abelian if:

  • It is preadditive, which means that for any two objects $X$ and $Y$ of $\mathcal{C}$, $\mathrm{Hom}(X,Y)$ has the structure of an abelian group. In a different language, this is the same as saying that $\mathcal{C}$ is enriched over $\text{Ab}$, the category of abelian groups.

  • It has a zero object $\mathbb{0}$, which is an object that is both initial and terminal in the category. Explicitly, for any object $X$ of $\mathcal{C}$, there is a unique morphism $\mathbb{0} \to X$, and there is a unique morphisms $X \to \mathbb{0}$.

  • The biproduct of any finite collection of objects in $\mathcal{C}$ exists. So for a collection $\{X_1,\dotsc,X_n\}$ of objects of $\mathcal{C}$, there is some object $\bigoplus_j X_j$ that is both the categorical product and coproduct of the $\{X_j\}_j$.

  • Every morphism has a kernel and a cokernel.

  • Lastly, every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism.

There are quite a few other equivalent characterizations of abelian categories too. These are typically the categories that we consider when studying homological algebra.

Examples & Conterexamples

  • For a ring $R$, the category $R\text{-Mod}$ of left $R$ modules is abelian. Each hom-set $\mathrm{Hom}(X,Y)$ of $R$-module homomorphisms inherits the structure of an abelian group from the abelian group structure on $Y$. The zero object of $R\text{-Mod}$ is the trivial module $\{0\}$ consisting of a single element. The biproduct of modules is usually called their direct sum. Any $R$-module homomorphisms admits a kernel and cokernel. And if you have a monomorphism $\varphi\colon N \hookrightarrow M$ then $\varphi$ is the kernel of the quotient map $M \to M/N$, and for an epimorphism $\psi\colon M \twoheadrightarrow L$ the map $\mathrm{Ker}\psi \to M$ will have cokernel $\psi$.

  • The category $\text{Ab}$ of abelian groups is an abelian category, since it is the module category $\mathbf{Z}\text{-Mod}$.

  • Given a quiver $Q$ and a field $\mathbf{k}$, the category $\mathrm{Rep}_{\mathbf{k}}Q$ of all $\mathbf{k}$-linear representations of $Q$ is an abelian category. This can be seen by directly showing that this category of representations is equivalent to a module category, the category of modules over the path algebra of $Q$.

Fundamental Theorems

The Freyd-Mitchell Embedding Theorem (nLab) is a precises statement of the catchphrase that abelian categories are "like" module categories.

Theorem — Every small abelian category admits a full, faithful, and exact functor to a category $R\text{-Mod}$ for some ring $R$.

Helpful Resources

824 questions
64
votes
1 answer

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 properties. It was an interest little shared by…
59
votes
3 answers

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 attempt to prove it. I believe I am missing some important…
56
votes
2 answers

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 very useful and efficient framework to work…
45
votes
1 answer

Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has a zero object, 3) $\mathcal{C}$ has binary…
Bruno Stonek
  • 11,791
  • 3
  • 54
  • 117
43
votes
4 answers

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' \mathrel{\overset{g'}{\to}} C' \to 0 $$ and morphisms $a : A \to…
40
votes
1 answer

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 $\mathcal{B}$. By the theory of derived functors, we…
39
votes
3 answers

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How can this be generalized to a general Abelian…
29
votes
1 answer

Has category theory solved major math problems?

I am new to category theory. Just wondering if category theory has solved any major problems for other mathematics fields? What are the major applications of category theory? Has anyone solved an open problem using category theory? Particularly,…
ben
  • 331
  • 2
  • 3
29
votes
1 answer

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 $D(A)=0$ and so it is abelian. Do you know what…
27
votes
2 answers

Meaning of "efface" in "effaceable functor" and "injective effacement"

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means to erase it, but I'm not sure if there's another…
25
votes
1 answer

Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, he states (without proof) that $\mathcal{A}$…
DBr
  • 4,460
  • 2
  • 27
  • 40
22
votes
1 answer

Category of quasicoherent sheaves not abelian

Wikipedia mentions that the category of quasicoherent sheaves need not form an abelian category on general ringed spaces. Is there a `naturally occurring' example of this failing, even for locally ringed spaces?
17
votes
1 answer

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact functor. Show that applying $F$ to an exact sequence…
Alan C
  • 1,852
  • 1
  • 15
  • 25
17
votes
2 answers

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 $f : X\rightarrow Y$ is an injective map of $A$-modules, we…
user15464
  • 11,042
  • 2
  • 37
  • 89
14
votes
4 answers

Homological algebra using nonabelian groups

Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice settings for homological algebra, but the notions of…
1
2 3
54 55