The minimum structure in which exact sequences make sense
An abelian category has a zero object, biproducts, kernels and cokernels, and satisfies the condition that every monomorphism is a kernel and every epimorphism a cokernel. Those axioms are exactly what is needed to define exactness, and everything in homological algebra up to derived functors goes through unchanged. The Freyd–Mitchell theorem then shows that any small abelian category embeds exactly into a module category, so element-based diagram chases remain legitimate.
Learning objectives
- State the axioms of an additive and an abelian category.
- Define exactness without reference to elements.
- List the standard examples beyond modules.
- State the Freyd–Mitchell theorem and what it licenses.
- Identify the Grothendieck axioms and when they are needed.
Section 01The axioms
- Stage 01AdditiveHom sets are abelian groups, composition is bilinear, there is a zero object and finite biproducts exist.
- Stage 02Kernels and cokernelsEvery morphism has a kernel and a cokernel.
- Stage 03NormalityEvery monomorphism is the kernel of its cokernel; every epimorphism is the cokernel of its kernel.
- Stage 04ConsequenceEvery morphism factors as an epi followed by a mono, and the factorisation is unique. Exactness can now be defined.
With the factorisation available, a sequence is exact at B when the image of the incoming morphism equals the kernel of the outgoing one — where image means the mono part of the factorisation.
Additive categories with kernels and cokernels are common; the requirement that monos and epis are normal is what rules out pathologies. In the category of topological abelian groups, for instance, a continuous injection with dense image is mono and epi but not iso, and normality fails.
Section 02Examples
| Category | Note |
|---|---|
| Modules over a ring | The motivating example; has enough projectives and injectives |
| Abelian groups | Modules over ℤ |
| Sheaves of modules on a space | Enough injectives but generally NOT enough projectives — the reason sheaf cohomology uses injective resolutions |
| Chain complexes in an abelian category | Abelian; the starting point for derived categories |
| Functors from a small category into an abelian category | Abelian pointwise |
| Finitely generated modules over a Noetherian ring | Abelian; not enough projectives in general |
| Grp | NOT abelian — not even additive |
Being abelian does not guarantee enough projectives or enough injectives. Sheaf categories are the standard case with injectives but no projectives, which is precisely why sheaf cohomology is defined by right derived functors of the global sections functor and has no left-derived counterpart.
Section 03The embedding theorem
Freyd–Mitchell: every small abelian category admits a full, faithful and exact embedding into a category of modules over some ring.
Licenses: proving a statement about finitely many objects and morphisms — any diagram lemma — by chasing elements in modules. The embedding is exact and faithful, so the conclusion transfers back. Does not license: assuming the category is a module category, using arbitrary products or colimits, or invoking projectives that exist in the target but not the source.
| Axiom | Requires |
|---|---|
| AB3 | Arbitrary coproducts exist |
| AB4 | Coproducts are exact |
| AB5 | Filtered colimits are exact |
| AB3*, AB4*, AB5* | The dual statements for products |
| Grothendieck category | AB5 plus a generator — guarantees enough injectives |
It is exactly the hypothesis under which injective resolutions can be constructed in general, which is what makes right derived functors available for sheaves. Module categories and sheaf categories both satisfy it.
ReferenceFrequently asked questions
Is the opposite of an abelian category abelian?
Yes — the axioms are self-dual. This is what makes the duality principle so powerful here: every theorem about abelian categories immediately yields its dual.
Can homological algebra be done without abelian categories?
Yes, and it is: exact categories, triangulated categories and higher-categorical frameworks all extend the reach. The abelian setting remains the natural home for Ext, Tor and derived functors in their classical form.
Do I need the embedding theorem to trust diagram lemmas?
Strictly, for element-based proofs, yes — or an intrinsic proof using generalised elements. In practice most texts prove the diagram lemmas by elements and cite the embedding once, which is legitimate provided the statement involves only a finite diagram.
NavigateContinue in this stream
Curated next steps from this page. The site also surfaces algorithmically related reading below.
ProvenanceSources and further reading
This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.