Mathematics.

homological algebra

Abelian Categories

Category Theory90 minDifficulty9 out of 10

You should know: categories and morphisms, functors, limits and colimits

Overview

An abelian category is a category in which you can do linear algebra and homological algebra: you can add morphisms, take kernels and cokernels, form exact sequences, and every short exact sequence behaves like 0 → A → B → C → 0. The category of abelian groups Ab, the category of R-modules R-Mod, and the category of sheaves of abelian groups on a space are all abelian. Grothendieck introduced the axioms in 1957 to unify these examples and enable cohomology for sheaves.

Intuition

An abelian category is a category where diagram-chasing proofs from linear algebra work verbatim. The Snake Lemma, Five Lemma, and long exact sequences in cohomology all follow from the axioms — you do not need to know the objects are modules. This is why sheaf cohomology and group cohomology can be treated uniformly.

Formal Definition

Definition

A category A is abelian if it satisfies:

Hom(A,B) is an abelian group, composition is bilinear\text{Hom}(A,B) \text{ is an abelian group, composition is bilinear}
Ab-enriched
A has a zero object 0 (both initial and terminal)\text{A has a zero object } 0 \text{ (both initial and terminal)}
zero-object
A has all finite products and coproducts (biproducts)\text{A has all finite products and coproducts (biproducts)}
biproducts
Every morphism has a kernel and a cokernel\text{Every morphism has a kernel and a cokernel}
ker-coker
Every monomorphism is a kernel; every epimorphism is a cokernel\text{Every monomorphism is a kernel; every epimorphism is a cokernel}
mono-epi

Properties

Self-duality

Aop is also abelian; kernels in Aop are cokernels in A.\mathcal{A}^{\text{op}} \text{ is also abelian; kernels in } \mathcal{A}^{\text{op}} \text{ are cokernels in } \mathcal{A}.

Canonical epi-mono factorisation

Every f:A→B factors as A↠im(f)↪B with coim(f)≅im(f).\text{Every } f: A \to B \text{ factors as } A \twoheadrightarrow \text{im}(f) \hookrightarrow B \text{ with } \text{coim}(f) \cong \text{im}(f).

Theorems

Theorem 1: Freyd–Mitchell Embedding Theorem
Every small abelian category A admits a full, exact, faithful embedding into R-Mod for some ring R.\text{Every small abelian category } \mathcal{A} \text{ admits a full, exact, faithful embedding into } R\text{-Mod for some ring } R.
Theorem 2: Snake Lemma
Given a commutative diagram with exact rows, there is an exact sequence ker⁡(a)→ker⁡(b)→ker⁡(c)→∂coker(a)→coker(b)→coker(c).\text{Given a commutative diagram with exact rows, there is an exact sequence } \ker(a) \to \ker(b) \to \ker(c) \xrightarrow{\partial} \text{coker}(a) \to \text{coker}(b) \to \text{coker}(c).
Theorem 3: Five Lemma
In a commutative diagram with exact rows and five vertical morphisms, if the outer four are isomorphisms, so is the middle one.\text{In a commutative diagram with exact rows and five vertical morphisms, if the outer four are isomorphisms, so is the middle one.}

Worked Examples

  1. Compute kernels: ker(a) = 0, ker(b) = 0, ker(c) = ker(0: ℤ/2ℤ → ℤ/3ℤ) = ℤ/2ℤ.

  2. Compute cokernels: coker(a) = 0, coker(b) = 0, coker(c) = ℤ/3ℤ.

  3. The Snake Lemma exact sequence is:

    0→0→0→Z/2Z→∂0→0→Z/3Z0 \to 0 \to 0 \to \mathbb{Z}/2\mathbb{Z} \xrightarrow{\partial} 0 \to 0 \to \mathbb{Z}/3\mathbb{Z}

Answer: The connecting map ∂: ℤ/2ℤ → 0 is zero; the sequence is exact at every term.

Practice Problems

Difficulty 7/10

In ℤ-Mod (= Ab), compute ker and coker of f: ℤ/6ℤ → ℤ/6ℤ given by f(x) = 2x.

Difficulty 8/10

Prove that in an abelian category, if f is both mono and epi, then f is an isomorphism.

Difficulty 9/10

State the Freyd–Mitchell embedding theorem and explain why it justifies 'element-chasing' proofs in any small abelian category.

Common Mistakes

Common Mistake

Thinking every category with kernels and cokernels is abelian.

An abelian category also requires the canonical map coim(f) → im(f) to be an isomorphism for every morphism f. This fails in the category of filtered abelian groups, for instance.

Common Mistake

Confusing epimorphisms with surjections in all abelian categories.

In Ab, epimorphisms are surjections. But in Sh(X, Ab), sheaf epimorphisms are surjections on stalks, not sectionwise — a common source of confusion in sheaf theory.

Quiz

Which of the following is NOT a requirement for an abelian category?
In an abelian category, the image of a morphism f: A → B is defined as:
The Freyd–Mitchell embedding theorem implies:

Historical Background

Grothendieck introduced the axioms of abelian categories in his landmark 1957 Tôhoku paper, motivated by algebraic geometry and sheaf theory. Freyd's embedding theorem (1964) showed that every small abelian category embeds fully faithfully into a module category.

  1. 1956

    Cartan–Eilenberg 'Homological Algebra' formalises exact sequences for modules

    Henri Cartan, Samuel Eilenberg

  2. 1957

    Grothendieck introduces abelian categories in the Tôhoku paper

    Alexander Grothendieck

  3. 1964

    Freyd–Mitchell embedding theorem proved

    Peter Freyd, Barry Mitchell

Summary

  • An abelian category is an Ab-enriched category with a zero object, finite biproducts, and kernels/cokernels for all morphisms.
  • The key axiom distinguishing abelian from additive: every mono is a kernel and every epi is a cokernel.
  • The Freyd–Mitchell theorem: every small abelian category embeds fully and exactly into R-Mod, justifying element-chasing proofs.
  • The Snake Lemma and Five Lemma hold in every abelian category, making abstract cohomology possible.
  • Key examples: Ab, R-Mod, Sh(X, Ab), and chain complexes in any abelian category.

References

  1. WebsiteWikipedia — Abelian category