Homological algebra possesses a peculiar character among mathematical disciplines. Its central arguments—the diagram chases, the long exact sequences, the spectral sequences cascading through filtrations—seem to work by magic. You place morphisms in certain configurations, invoke exactness at various points, and conclusions emerge as if from pure diagram manipulation. Yet beneath this apparent sorcery lies a carefully engineered categorical structure that makes such reasoning valid.

This structure is the abelian category. When Grothendieck and others formalized this concept in the mid-twentieth century, they were answering a fundamental question: what minimal categorical apparatus suffices for homological algebra to function? The answer revealed that surprisingly few axioms—concerning kernels, cokernels, and their interplay—generate the entire machinery of exact sequences and diagram chasing. Categories of modules, sheaves of abelian groups, and chain complexes all satisfy these axioms, explaining why homological techniques transfer seamlessly between these seemingly different contexts.

Understanding abelian categories means understanding why certain categorical environments support a particular mode of mathematical reasoning. It requires grasping how the existence of kernels and cokernels, combined with a crucial factorization property, creates an arena where information flow through sequences can be precisely tracked. The snake lemma, perhaps the most celebrated diagram chase, emerges not as a trick but as an inevitable consequence of this structure—a theorem that could not fail to be true once the categorical axioms are in place.

In an abelian category, every morphism f: A → B possesses both a kernel and a cokernel, and these constructions exhibit properties far stronger than their mere existence suggests. The kernel ker(f) → A is the categorical representation of 'elements mapped to zero'—it is the universal morphism through which any other morphism that composes with f to give zero must factor. Dually, the cokernel B → coker(f) captures 'quotienting by the image,' universally receiving any morphism from B that annihilates everything coming from A.

What distinguishes abelian categories from merely additive categories with kernels and cokernels is the image-coimage factorization. Every morphism f factors as A → coim(f) → im(f) → B, where the coimage is the cokernel of the kernel, and the image is the kernel of the cokernel. The crucial abelian axiom demands that the induced map coim(f) → im(f) is an isomorphism. This seemingly technical condition encodes a profound fact: the two natural ways of measuring 'what f actually does'—quotienting the domain by what goes to zero, versus identifying the target points actually hit—yield the same answer.

Consider how this plays out in the category of abelian groups. For a homomorphism f: A → B, the kernel is the subgroup of elements sent to zero, while the cokernel is B/f(A). The coimage A/ker(f) captures 'A with redundancies removed,' and the first isomorphism theorem states precisely that this equals the image f(A). The abelian axiom thus generalizes the first isomorphism theorem to an arbitrary categorical setting.

This factorization enables a crucial distinction: a morphism is a monomorphism (injective in spirit) precisely when its kernel is zero, and an epimorphism (surjective in spirit) precisely when its cokernel is zero. Moreover, in abelian categories, these notions coincide with the categorical definitions of mono and epi, a coincidence that fails in many other contexts. The kernel and cokernel become diagnostic tools for understanding the behavior of morphisms.

The interplay between kernels and cokernels generates the basic vocabulary of homological algebra. Short exact sequences 0 → A → B → C → 0 express that A is the kernel of B → C while C is its cokernel. This compression of information—stating simultaneously that A embeds, C quotients, and the embedding captures exactly what the quotient kills—becomes expressible precisely because our category possesses the requisite universal constructions with the right properties.

Takeaway

The image-coimage isomorphism is the abelian axiom that transforms categories with kernels and cokernels into arenas where homological algebra can function—it ensures that the two natural measures of a morphism's behavior coincide.

A sequence of morphisms A → B → C in an abelian category is exact at B when the image of the first morphism equals the kernel of the second. This definition, simple as it appears, encodes a principle of information conservation: whatever enters B from A is precisely what gets annihilated when moving to C. Nothing is lost in transit; nothing new appears to be killed.

Exactness extends to longer sequences: A₀ → A₁ → A₂ → ... is exact when it is exact at each intermediate object. The paradigmatic example is the short exact sequence 0 → A → B → C → 0, exact at all five positions. Exactness at A says the map from 0 → A has zero kernel, meaning the map A → B is a monomorphism. Exactness at C says the map C → 0 has B → C as its kernel, meaning B → C is an epimorphism. Exactness at B itself forces A to be the kernel of B → C.

The power of exact sequences lies in their functorial behavior. When you apply a functor to an exact sequence, exactness may or may not be preserved. Left exact functors preserve exactness at the left end; right exact functors at the right end. Derived functors—the central construction of homological algebra—measure precisely the failure of exactness under functor application, transforming short exact sequences into long exact sequences that track this failure through infinite chains of derived objects.

Understanding exactness requires appreciating what it does not say. An exact sequence A → B → C guarantees that im(A → B) = ker(B → C), but this common object might be 'small' or 'large' relative to B. The sequence 0 → ℤ → ℤ → ℤ/2ℤ → 0, with the first map being multiplication by 2, and the second being projection, is exact. But the sequence ℤ → ℤ → 0, with the first map again multiplication by 2, is not—the image is 2ℤ while the kernel of ℤ → 0 is all of ℤ.

Exact sequences thus function as structural equations. Just as algebraic equations constrain numerical relationships, exact sequences constrain morphism relationships. Solving homological problems often means constructing exact sequences that translate difficult questions about objects into manageable questions about morphisms and their kernels. The information encoded in exactness—that images match kernels throughout—becomes the currency of computation.

Takeaway

Exactness captures the principle that information flows through a sequence without loss or spontaneous generation—what exits one object as image enters the next as precisely what will be killed.

The snake lemma stands as the exemplar of diagram chasing, demonstrating how categorical structure enables computational techniques of surprising power. Consider a commutative diagram with exact rows: two short exact sequences 0 → A → B → C → 0 and 0 → A' → B' → C' → 0, connected by vertical morphisms f: A → A', g: B → B', h: C → C'. The snake lemma asserts the existence of an exact sequence ker(f) → ker(g) → ker(h) → coker(f) → coker(g) → coker(h), where the connecting morphism ker(h) → coker(f) is the 'snake' winding through the diagram.

The proof proceeds by diagram chasing—an element-based argument that nevertheless works in any abelian category through the embedding theorem. This theorem guarantees that every small abelian category embeds fully and faithfully into a category of modules, where element-based arguments are legitimate. Thus we can 'pretend' our objects have elements, chase them through the diagram, and know the resulting construction is valid categorically.

To construct the connecting morphism, start with an element in ker(h)—something in C killed by h. By exactness of the top row, this lifts to some element of B. Apply g to get an element of B'. By commutativity, this maps to zero in C'. By exactness of the bottom row, our element in B' comes from something in A'. This something is well-defined up to the image of f, hence determines an element of coker(f). This winding path—up from C, across through B, down to B', back through A'—traces the snake through the diagram.

Verifying exactness of the resulting sequence requires further diagram chases, each exploiting the interplay of the various exactness hypotheses. The snake lemma's proof thus demonstrates a method: given categorical structure (kernels, cokernels, exact sequences), one can reason about morphisms by tracking 'elements' through diagrams, using universality to ensure constructions are well-defined and naturality to ensure compatibilities.

The snake lemma's importance transcends its specific statement. It generates connecting morphisms in long exact sequences of derived functors, in the long exact sequence of a pair in homology, in the boundary maps of spectral sequences. Each of these fundamental constructions is, at its core, an application of the snake lemma to an appropriate diagram. The categorical structure of abelian categories makes such applications automatic—once you have the diagram with exact rows, the snake emerges inevitably.

Takeaway

The snake lemma is not a clever trick but an inevitable consequence of abelian structure—given the right diagram, the connecting morphism must exist, and the resulting sequence must be exact.

Abelian categories reveal that homological algebra's power stems not from particular objects—modules, sheaves, chain complexes—but from structural properties these categories share. The existence of kernels and cokernels, the image-coimage isomorphism, and the resulting calculus of exact sequences create an environment where diagram chasing becomes rigorous mathematics rather than hopeful manipulation.

This perspective transforms how we understand classical results. The snake lemma, the five lemma, the nine lemma—these cease to be theorems requiring individual proofs and become manifestations of a single phenomenon: the precise manner in which information propagates through exact sequences in abelian categories. Understanding the categorical structure means understanding why all these results must hold simultaneously.

For the working mathematician, abelian categories offer both economy and power. Economy, because a single theory applies uniformly across diverse concrete settings. Power, because the abstract framework clarifies which properties matter and suggests generalizations—to derived categories, triangulated categories, and beyond—where new forms of homological algebra become possible.