Abelian Categories and Exact Sequences

⚡ Key Takeaways

An abelian category is a category that behaves like the category of modules over a ring: it has a zero object, all finite products and coproducts, and — crucially — every morphism has a kernel and cokernel such that monomorphisms are kernels and epimorphisms are cokernels. Exact sequences are the central computational and structural tool: chains of objects and morphisms where the image of each map equals the kernel of the next. Together they form the categorical foundation of homological algebra, applicable across algebra, topology, algebraic geometry, and representation theory. Introduced by Buchsbaum (1955) and Grothendieck (1957) in his Tôhoku paper.

💡 The Core Insight

In an abelian category, every morphism factors canonically as an epimorphism followed by a monomorphism, $f = (\text{mono}) \circ (\text{epi})$ , with the middle object being the image — and this factorization is unique up to unique isomorphism. This single fact is what makes the notion of an exact sequence well-defined and what allows the entire machinery of homology (the failure of exactness, measured as $\ker / \operatorname{im}$ ) to be formulated purely categorically.

DefinitionAb-category (preadditive category)

A category $\mathcal{A}$ is an Ab-category if every hom-set $\operatorname{Hom}_{\mathcal{A}}(X,Y)$ carries the structure of an abelian group, and composition is bilinear:

1

\quad (f+f')\circ g = f\circ g + f'\circ g, \qquad h\circ(f+f') = h\circ f + h\circ f'.

Derivation hint: This is the requirement that composition $\circ:\operatorname{Hom}(Y,Z)\times\operatorname{Hom}(X,Y)\to\operatorname{Hom}(X,Z)$ be $\mathbb{Z}$ -bilinear, i.e., a group homomorphism in each variable. Here $f,f'\in\operatorname{Hom}(X,Y)$ , $g\in\operatorname{Hom}(W,X)$ , $h\in\operatorname{Hom}(Y,Z)$ . The additive identity is the zero morphism $0_{XY}$ .

DefinitionAdditive category

An Ab-category $\mathcal{A}$ is additive if it has a zero object $0$ (an object that is both initial and terminal) and all finite biproducts. The biproduct $X\oplus Y$ comes with injections $i_X, i_Y$ and projections $p_X, p_Y$ satisfying:

2

\quad p_X i_X = 1_X,\quad p_Y i_Y = 1_Y,\quad p_X i_Y = 0,\quad p_Y i_X = 0,\quad i_X p_X + i_Y p_Y = 1_{X\oplus Y}.

Derivation hint: The first four equations characterize $X\oplus Y$ as both product and coproduct; the last (completeness relation) forces the product and coproduct to coincide, which is automatic in an Ab-category.

DefinitionPreabelian and abelian categories

A preabelian category is an additive category in which every morphism $f:X\to Y$ has a kernel $\ker f \xrightarrow{k} X$ and a cokernel $Y \xrightarrow{c} \operatorname{coker} f$ , characterized by the universal properties:

3

\quad f\circ k = 0,\ \text{and } \forall\, u\ (f u=0 \Rightarrow \exists!\, \bar u:\, k\bar u = u);\quad c\circ f = 0,\ \text{and } \forall\, v\ (vf=0 \Rightarrow \exists!\, \bar v:\, \bar v c = v).

Derivation hint: A kernel is the equalizer of $(f,0)$ ; a cokernel is the coequalizer of $(f,0)$ . Universal property = unique factorization through any competing map killing $f$ (resp. killed by $f$ ). An additive category $\mathcal{A}$ is abelian if it is preabelian and, in addition, every monomorphism is a kernel (of its cokernel) and every epimorphism is a cokernel (of its kernel):

4

\quad m \text{ mono} \implies m = \ker(\operatorname{coker} m), \qquad e \text{ epi} \implies e = \operatorname{coker}(\ker e).

Derivation hint: These "normality" conditions are exactly what fail in non-abelian additive categories (e.g. filtered abelian groups, or torsion-free abelian groups); they guarantee the image–coimage comparison map is an isomorphism.

TheoremCanonical image factorization

In an abelian category, every morphism $f:X\to Y$ admits a factorization

5

\quad X \xrightarrow{\;e\;} \operatorname{im} f \xrightarrow{\;m\;} Y, \qquad f = m\circ e,

where $e$ is an epimorphism and $m$ is a monomorphism, and $\operatorname{im} f := \ker(\operatorname{coker} f)$ . The coimage $\operatorname{coim} f := \operatorname{coker}(\ker f)$ , and the canonical comparison map

6

\quad \theta:\ \operatorname{coim} f \xrightarrow{\ \sim\ } \operatorname{im} f

Derivation hint: $\theta$ exists in any preabelian category by universal properties; abelianness ( $[4]$ ) is precisely the requirement that $\theta$ be an isomorphism for all $f$ . is an isomorphism. This factorization is unique up to unique isomorphism.

DefinitionExact sequence

A pair of composable morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ with $gf=0$ is exact at $Y$ if the canonical monomorphism $\operatorname{im} f \hookrightarrow Y$ equals $\ker g$ as subobjects of $Y$ :

7

\quad \operatorname{im} f = \ker g \quad\text{(as subobjects of } Y\text{)}.

Derivation hint: $gf=0$ already gives $\operatorname{im} f\subseteq\ker g$ ; exactness is the reverse inclusion. The "homology object" $H = \ker g / \operatorname{im} f := \operatorname{coker}(\operatorname{im} f \to \ker g)$ measures the failure of exactness. A short exact sequence is an exact sequence

8

\quad 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0,

Derivation hint: Exactness at $A$ means $f$ is mono; at $C$ means $g$ is epi; at $B$ means $\operatorname{im} f=\ker g$ . Equivalently $A=\ker g$ and $C=\operatorname{coker} f$ . equivalently: $f$ is a monomorphism, $g$ is an epimorphism, and $f = \ker g$ (hence $g = \operatorname{coker} f$ ).

Figure

Canonical epi–mono factorization and the comparison map

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ProofExistence and uniqueness of the image factorization

Step 1 (Construct the cokernel and its kernel). Given $f:X\to Y$ , form $c=\operatorname{coker} f:Y\to Q$ . Since $\mathcal{A}$ is preabelian, $c$ has a kernel $m:K\to Y$ . Define $\operatorname{im} f := K$ and the monomorphism is $m$ .

Step 2 ( $f$ factors through $m$ ). We have $cf = 0$ by definition of cokernel. By the universal property of $k=\ker c$ ( $[3]$ ), there is a unique $e:X\to K$ with $m e = f$ . This gives $f = m\circ e$ .

Step 3 ( $m$ is monic). Kernels are always monomorphisms: if $m u = m u'$ then $m(u-u')=0$ , and since $m$ equalizes nothing but $0$ into $Y$ killed by $c$ ... more directly, $m=\ker c$ is by construction the equalizer of $(c,0)$ , and equalizers are monic.

Step 4 ( $e$ is epic) — this is where abelianness enters. In a general preabelian category $e$ need not be epi. By axiom $[4]$ , the dual construction gives $\operatorname{coim} f = \operatorname{coker}(\ker f)$ with the universal epimorphism $e':X\to\operatorname{coim} f$ , and a unique monomorphism $\operatorname{coim} f \to Y$ factoring $f$ . The comparison map $\theta:\operatorname{coim} f\to\operatorname{im} f$ satisfies $\theta e' = e$ (after identifications). Abelianness forces $\theta$ to be an isomorphism (this is the content of every mono being a kernel and every epi a cokernel), hence $e = \theta e'$ is epi because $e'$ is.

Step 5 (Uniqueness). Suppose $f = m''e''$ with $m''$ mono, $e''$ epi. Since $m''$ is mono it equals $\ker(\operatorname{coker} m'')$ ; one shows $\operatorname{coker} f = \operatorname{coker} m''$ , hence $m''$ and $m$ have the same image, giving a unique iso $\operatorname{im} f \cong \operatorname{im}'' f$ commuting with everything. $\blacksquare$

∎

ProofThe Snake Lemma (statement and key steps)

Given a commutative diagram with exact rows in an abelian category

9

\quad \begin{array}{ccccccccc} & & A & \xrightarrow{} & B & \xrightarrow{} & C & \to & 0\\ & & \downarrow a & & \downarrow b & & \downarrow c & & \\ 0 & \to & A' & \xrightarrow{} & B' & \xrightarrow{} & C' & & \end{array}

Derivation hint: Rows exact means top is exact at $A,B$ and bottom at $B',C'$ ; columns are the three vertical maps $a,b,c$ . there is a natural connecting homomorphism $\delta:\ker c \to \operatorname{coker} a$ and a long exact sequence

10

\quad \ker a \to \ker b \to \ker c \xrightarrow{\;\delta\;} \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c.

Derivation hint: $\delta$ is built by "lifting": take $x\in\ker c$ , pull back to $B$ , push down by $b$ into $B'$ , then it lands in $\operatorname{im}(A'\to B')\cong A'$ , project to $\operatorname{coker} a$ . Each choice is well-defined modulo the image of $a$ .

Construction of $\delta$ (element-style, justified by Freyd–Mitchell): Let $x\in\ker c\subseteq C$ . Choose $\tilde x\in B$ with $g(\tilde x)=x$ (possible as $g$ epi). Then $g'(b\tilde x)=c(g\tilde x)=c(x)=0$ , so $b\tilde x\in\ker g'=\operatorname{im} f'$ ; write $b\tilde x = f'(y)$ for unique $y\in A'$ . Set $\delta(x):=[y]\in\operatorname{coker} a$ . Independence of the choice of $\tilde x$ follows because two lifts differ by an element of $\operatorname{im} f = \ker g$ , whose image under $b$ lands in $f'(\operatorname{im} a)$ . Exactness at each spot is verified by chasing. $\blacksquare$

∎

mathematician · Alexander Grothendieck | 1928–2014

In the 1957 Tôhoku paper "Sur quelques points d'algèbre homologique," Grothendieck recast homological algebra around abelian categories, introducing the axioms AB1–AB6 and AB3\–AB5\ governing existence and exactness of limits and colimits. His key innovations were the recognition that injective objects suffice for derived functors (the AB5 + a generator conditions ensuring enough injectives — the Grothendieck category concept) and the systematic use of these structures to compute sheaf cohomology. This abstraction was driven by the needs of algebraic geometry, where sheaves of modules form an abelian category that is not a module category, yet must support all homological machinery. Grothendieck's framework became the language of EGA/SGA and of essentially all modern cohomology theories.

🧠 Intuition

Think of an abelian category as " $R\text{-}\mathbf{Mod}$ stripped to its skeleton." Everything you do with modules — kernels, images, quotients, direct sums, short exact sequences, diagram chases — survives, but you forget what the elements are. The two normality axioms ( $[4]$ ) are doing all the work: they say subobjects and quotient objects are dual and complete. In a generic additive category you can have a mono that is not the kernel of anything (e.g. $\mathbb{Z}\xrightarrow{2}\mathbb{Z}$ inside torsion-free groups behaves fine, but consider the inclusion of a non-closed subspace), and then "the quotient" loses information about how the subobject sits inside. Abelianness restores the perfect lattice-theoretic duality: the poset of subobjects of $X$ is anti-isomorphic to the poset of quotient objects of $X$ , via $S\mapsto X/S$ . Exactness is then simply the statement "what dies next is exactly what was born here," and homology is the bookkeeping of the discrepancy.

ExampleComputing homology of a complex of abelian groups

Given: the complex of $\mathbb{Z}$ -modules

\mathbb{Z} \xrightarrow{\;d_2=\binom{2}{0}\;} \mathbb{Z}^2 \xrightarrow{\;d_1=(0\ \ 3)\;} \mathbb{Z}.

First check $d_1 d_2 = (0\ \ 3)\binom{2}{0} = 0$ . ✓ It is a complex.

Step 1 — compute at $\mathbb{Z}^2$ . $\ker d_1 = \{(a,b): 3b=0\} = \{(a,0)\} \cong \mathbb{Z}$ (the first factor). $\operatorname{im} d_2 = \{(2k,0): k\in\mathbb{Z}\} = 2\mathbb{Z}\oplus 0$ .

Step 2 — homology.

H_1 = \ker d_1 / \operatorname{im} d_2 \cong \mathbb{Z}/2\mathbb{Z}.

Step 3 — at the right $\mathbb{Z}$ . $\ker(0) = \mathbb{Z}$ , $\operatorname{im} d_1 = 3\mathbb{Z}$ , so $H_0 = \mathbb{Z}/3\mathbb{Z}$ .

Boxed answer:

\boxed{H_1 \cong \mathbb{Z}/2\mathbb{Z}, \qquad H_0 \cong \mathbb{Z}/3\mathbb{Z}, \qquad H_2 = \ker d_2 = 0.}

The nonzero homology measures the failure of exactness exactly at the two spots.

ExampleThe connecting map and a Tor computation via the snake lemma

Given: the short exact sequence of $\mathbb{Z}$ -modules $0\to\mathbb{Z}\xrightarrow{n}\mathbb{Z}\to\mathbb{Z}/n\to 0$ . Apply $-\otimes_{\mathbb{Z}} M$ for $M=\mathbb{Z}/m$ , with $d=\gcd(m,n)$ .

Step 1 — tensor (right exact). We get

\mathbb{Z}/m \xrightarrow{\;\cdot n\;} \mathbb{Z}/m \to \mathbb{Z}/n\otimes\mathbb{Z}/m \to 0.

Step 2 — identify $\operatorname{coker}$ . Multiplication by $n$ on $\mathbb{Z}/m$ has image $n(\mathbb{Z}/m) = d(\mathbb{Z}/m)$ , so

\mathbb{Z}/n\otimes\mathbb{Z}/m \cong (\mathbb{Z}/m)/(d\mathbb{Z}/m) \cong \mathbb{Z}/d.

Step 3 — kernel gives Tor. $\ker(\cdot n:\mathbb{Z}/m\to\mathbb{Z}/m) = \{x : nx\equiv 0 \pmod m\}$ has order $d$ , so

\operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/m)\cong\mathbb{Z}/d.

Boxed answer (take $m=6,n=4$ , $d=2$ ):

\boxed{\mathbb{Z}/4\otimes\mathbb{Z}/6 \cong \mathbb{Z}/2, \qquad \operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/4,\mathbb{Z}/6)\cong\mathbb{Z}/2.}

This is the long exact sequence of $\operatorname{Tor}$ in action — the failure of $-\otimes M$ to be left exact is captured categorically.

⚠️ Common mistake

"Mono + epi = iso." In an arbitrary category this is false, and even in many additive categories it fails. In an abelian category it is true: a morphism that is both monic and epic is an isomorphism (because $f$ mono $\Rightarrow \ker f=0 \Rightarrow \operatorname{coim} f = X$ ; $f$ epi $\Rightarrow \operatorname{coker} f=0 \Rightarrow \operatorname{im} f = Y$ ; the comparison iso $\theta$ then makes $f$ an iso). But beware: this relies critically on abelianness. In $\mathbf{Top}$ or $\mathbf{Ring}$ a bimorphism need not be an iso (e.g. $\mathbb{Z}\hookrightarrow\mathbb{Q}$ is mono and epi in $\mathbf{Ring}$ but not surjective).

⚠️ Common mistake

Exact does not mean split. A short exact sequence $0\to A\to B\to C\to 0$ being exact only says $B$ is an extension of $C$ by $A$ ; it need not split, i.e. $B\not\cong A\oplus C$ in general. The sequence $0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$ does not split. Splitting is governed by $\operatorname{Ext}^1(C,A)$ ; the sequence splits iff its class in $\operatorname{Ext}^1$ is zero.

CounterexampleAn additive category that is NOT abelian

Consider $\mathcal{A} =$ the category of finitely generated free abelian groups (equivalently, finite-rank lattices) with group homomorphisms. It is additive: it has $0$ , biproducts $\mathbb{Z}^m\oplus\mathbb{Z}^n=\mathbb{Z}^{m+n}$ , and additive hom-sets. But it is not abelian. The morphism $\cdot 2:\mathbb{Z}\to\mathbb{Z}$ is a monomorphism whose cokernel $\mathbb{Z}/2$ does not exist in $\mathcal{A}$ (it's not free) — so kernels/cokernels fail to exist, violating even the preabelian requirement. Even fixing existence, the category of torsion-free abelian groups is preabelian but not abelian: there the mono $\cdot 2:\mathbb{Z}\to\mathbb{Z}$ has cokernel $0$ (since $\mathbb{Z}/2$ is killed by the torsion-free reflection), so $\ker(\operatorname{coker}(\cdot 2)) = \ker(0\text{-map}) = \mathbb{Z} \ne \operatorname{im}(\cdot2)=2\mathbb{Z}$ . The comparison map $\theta$ is the mono $2\mathbb{Z}\hookrightarrow\mathbb{Z}$ , not an iso — axiom $[4]$ fails.

connection

Abelian categories sit at the crossroads of nearly all homological mathematics. The Freyd–Mitchell Embedding Theorem legitimizes element-wise diagram chasing by embedding any small abelian category into $R\text{-}\mathbf{Mod}$ . The notion of exactness powers Derived Functors ( $\operatorname{Ext}$ , $\operatorname{Tor}$ , sheaf cohomology) and the formalism of Chain Complexes and Homology. Refining the structure leads to Triangulated Categories and Derived Categories, which are obtained by localizing complexes over an abelian category at quasi-isomorphisms. The Yoneda Ext Groups interpret $\operatorname{Ext}^n$ as equivalence classes of $n$ -fold extensions, a purely abelian-categorical construction. Finally, Grothendieck Categories (AB5 + a generator) are exactly the abelian categories with enough injectives where Grothendieck's derived-functor cohomology lives, and Serre Subcategories and Quotient Categories generalize localization of rings to this setting.

🔬 Where this appears

⏳ History

- 1942–45 — Eilenberg and Mac Lane introduce category theory and natural transformations, originally to formalize naturality in homology/cohomology of topological spaces.

- 1945 — Eilenberg–Steenrod axiomatize homology theory, foregrounding exact sequences as the central structural device.

- 1952 — Eilenberg–Steenrod's Foundations of Algebraic Topology and Cartan–Eilenberg's developing work crystallize "homological algebra" as a subject over module categories.

- 1955 — David Buchsbaum, in his thesis and the appendix to Cartan–Eilenberg, gives the first axioms for "exact categories," essentially the abelian category concept.

- 1956 — Cartan–Eilenberg's Homological Algebra published, the definitive treatment over rings, exposing the need for a categorical generalization for sheaves.

- 1957 — Grothendieck's Tôhoku paper defines abelian categories with axioms AB1–AB6, introduces Grothendieck categories, enough injectives, and recasts sheaf cohomology as derived functors — the watershed.

- 1960s — Grothendieck's EGA/SGA deploy abelian categories pervasively; the derived category is introduced by Verdier (Grothendieck's student) in his thesis.

- 1964 — Peter Freyd (Abelian Categories) and Barry Mitchell (1965, Theory of Categories) prove the full embedding theorem, justifying element-wise reasoning.

- 1973–82 — Beilinson–Bernstein–Deligne–Gabber develop t-structures and perverse sheaves, identifying new abelian categories as hearts inside triangulated/derived categories.

open-problem

Where the abelian framework strains. (i) The derived category of an abelian category is not itself abelian — it is only triangulated; the desire for a single homotopy-coherent enhancement led to stable $\infty$ -categories and dg-categories, where the precise comparison and the existence of "enhancements" (Bondal–Larsen–Lunts, Lunts–Orlov) involve deep, still-active questions. (ii) Non-additive and "exact"-but-not-abelian settings: many categories (vector bundles, filtered modules, Banach spaces) are naturally quasi-abelian or merely exact categories (Quillen), where short exact sequences are specified extra data; classifying and relating these to abelian categories (the derived category of a quasi-abelian category, Schneiders) remains subtle. (iii) The Freyd–Mitchell theorem requires smallness; for large abelian categories (e.g. all modules over a proper class of rings) a clean concrete embedding is delicate. (iv) Tannakian reconstruction: precisely which abelian (tensor) categories arise as representation categories of (pro)algebraic groups is governed by deep criteria (Deligne's tensor-category theorems), and the non-neutral and characteristic- $p$ cases retain open structural questions.

Abelian Categories and Exact Sequences

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also