Abelian category

1 Basic definitions and axioms

1.1 Additive category requirements

An Abelian category is built on top of an additive category. In an additive category, morphism sets carry an internal notion of addition compatible with composition, and finite products and finite coproducts agree in a way that produces “direct sum–like” behavior. This additive structure is the minimal categorical framework in which expressions such as “image,” “kernel,” and “cokernel” can be formulated cleanly.

Concretely, an additive category has a zero morphism between any two objects, admits biproducts for any pair of objects, and every hom-set is equipped with an abelian group structure such that composition is bilinear.

1.2 Kernels and cokernels

Within an additive category, a kernel of a morphism \(f: A \to B\) is an object \(\ker(f)\) equipped with a morphism \(\iota:\ker(f)\to A\) such that \(f\circ \iota=0\) and \(\iota\) is universal among morphisms into \(A\) that become zero after applying \(f\). Dually, a cokernel of \(f\) is an object \(\operatorname{coker}(f)\) with a morphism \(\pi:B\to \operatorname{coker}(f)\) such that \(\pi\circ f=0\) and \(\pi\) is universal among maps out of \(B\) that annihilate \(f\).

An Abelian category requires that every morphism has both a kernel and a cokernel.

1.3 Monomorphisms, epimorphisms, and normality

A morphism is a monomorphism if it is left-cancellable, and an epimorphism if it is right-cancellable, in the categorical sense. In many categories these notions differ from kernel/cokernel notions; in an Abelian category they align strongly.

“Normality” refers to the requirement that monomorphisms and epimorphisms arise from kernels and cokernels: specifically, every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism. Together with the existence of kernels and cokernels, these conditions give the characteristic stability of Abelian categories.

1.4 Exactness in an Abelian category

Given morphisms \(A\xrightarrow{f}B\xrightarrow{g}C\), one says the sequence \[ A \xrightarrow{f} B \xrightarrow{g} C \] is exact at \(B\) if \(\operatorname{im}(f)\) (defined via kernels/cokernels) equals \(\ker(g)\). In an Abelian category, this notion is equivalent to the statement that \(f\) is the kernel of \(g\) modulo the appropriate identifications, and dually that \(g\) is a cokernel of \(f\). Exactness is therefore phrased in a way that does not depend on additional structure beyond kernels and cokernels.

Because kernels and cokernels behave predictably, exactness becomes a robust tool for building long sequences of invariants.

1.5 Zero object and biproducts

An additive setting features a distinguished object \(0\) that is both initial and terminal, and hence provides canonical “zero maps.” For objects \(A\) and \(B\), the Abelian category also has biproducts \(A\oplus B\), meaning there are canonical inclusion and projection maps making \(A\oplus B\) simultaneously a product and a coproduct. This property supports linear algebra–style manipulations within category theory.

2 Additive structure and categorical arithmetic

2.1 Hom-sets as abelian groups

The hallmark of additivity is that each hom-set \(\operatorname{Hom}(A,B)\) is an abelian group. Morphism addition and negation are compatible with composition: composing with a fixed morphism on either side is a group homomorphism on hom-sets. This turns categorical statements involving linear combinations into well-defined algebraic assertions.

As a result, constructions like “difference of morphisms” and “bilinear pairing with composition” are internally meaningful.

2.2 Zero morphisms and bilinearity

In an additive category, the zero morphism \(0_{A,B}\) is characterized by being the additive identity in \(\operatorname{Hom}(A,B)\). Bilinearity means that for morphisms \(f,f':A\to B\) and \(g:B\to C\), and similarly for composition on the other side, one has: \[ g\circ (f+f') = g\circ f + g\circ f' \quad\text{and}\quad (g+g')\circ f = g\circ f + g'\circ f. \] This algebraic control is what enables exactness criteria to interact smoothly with diagrams.

2.3 Biproducts vs. direct sums

Although \(A\oplus B\) is described categorically as a biproduct, it behaves like the direct sum from linear algebra and module theory. The biproduct structure supplies canonical maps \[ A \xrightarrow{i_A} A\oplus B \xrightarrow{p_A} A, \qquad B \xrightarrow{i_B} A\oplus B \xrightarrow{p_B} B \] with \(p_A i_A = \mathrm{id}_A\), \(p_B i_B=\mathrm{id}_B\), and cross compositions \(p_A i_B\) and \(p_B i_A\) equal zero. This makes \(A\oplus B\) a linear container with independent components.

2.4 Idempotents and splitting (overview)

Idempotent endomorphisms \(e:A\to A\) with \(e^2=e\) generalize projections. A category is said to have splitting of idempotents if every such \(e\) comes from an actual decomposition of \(A\) into a direct summand. Abelian categories have strong properties in this regard, enabling decomposition arguments and the construction of direct summands from algebraic data.

(While the precise “splitting” results are often developed in terms of exactness and abelian axioms, the guiding intuition is that idempotents act like projections onto summands.)

3 Exact sequences and homological notions

3.1 Short exact sequences

A short exact sequence is a chain \[ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 \] such that \(f\) is a kernel of \(g\) and \(g\) is a cokernel of \(f\). In an Abelian category this is equivalent to saying the image of \(f\) equals the kernel of \(g\), and the cokernel of \(f\) identifies with \(C\). Short exact sequences express how an object \(B\) is assembled from \(A\) and \(C\) in a controlled way.

They serve as the basic units for defining derived invariants and for inductive arguments.

3.2 Long exact sequences (setup and examples)

Long exact sequences arise when one applies a (co)variant functor to a short exact sequence and uses exactness properties to obtain an extended chain of groups or objects. In homological algebra, applying homology, Ext, or Tor functors to short exact sequences produces connecting morphisms and yields long exact sequences.

The key structural input is that kernels and cokernels can be tracked functorially, so the failure of a functor to preserve exactness manifests through canonical connecting maps.

3.3 Split exact sequences

A short exact sequence is split if \(B\) is isomorphic to \(A\oplus C\) in a way compatible with the maps. Equivalently, the sequence becomes exact in a stronger sense where the inclusion \(A\to B\) admits a retraction or the projection \(B\to C\) admits a section. Split exact sequences are “trivial” extensions, and their study clarifies which short exact sequences represent genuinely new gluing data.

3.4 Extensions and extension classes

Extensions of \(C\) by \(A\) describe equivalence classes of short exact sequences \[ 0 \to A \to B \to C \to 0. \] Two extensions are considered equivalent if there is an isomorphism between the middle terms making the outer terms correspond. In many settings, extension groups (notably Ext) classify these classes, and the set of extensions inherits algebraic operations compatible with Baer sum constructions.

This viewpoint turns extension problems into computable invariants.

4 Functors between Abelian categories

4.1 Exact functors

A functor \(F:\mathcal{A}\to\mathcal{B}\) between Abelian categories is exact if it preserves short exact sequences. Equivalently, it preserves kernels and cokernels in the sense that the image of an exact sequence is exact at every stage. Exact functors are the categorical analogs of “linear” transformations that respect the intrinsic exact structure.

4.2 Left exact and right exact functors

A functor is left exact if it preserves kernels (and thus preserves exactness at the left) and right exact if it preserves cokernels. This split of requirements is central in homological algebra: many natural constructions fail to be fully exact, but they preserve one side of exactness. The non-exactness is then captured by derived functors.

4.3 Additive functors and preservation of limits/colimits

In Abelian categories, most homological functors are additive. An additive functor respects the abelian group structure on hom-sets and therefore interacts well with biproducts. Preservation of limits and colimits ties exactness to the behavior of products and coproducts: kernel-like behavior aligns with left exactness, while cokernel-like behavior aligns with right exactness.

4.4 Faithful, full, and conservative functors (relations to exactness)

A faithful functor reflects injectivity properties of morphisms but not necessarily exactness. A full functor reflects the existence of morphisms in a stronger sense, while a conservative functor reflects isomorphisms. These properties relate indirectly to exactness: exactness is about preservation of specific categorical structures (kernels/cokernels), and conservative or fully faithful functors can sometimes allow one to transport exactness assertions back and forth—provided the functor also interacts suitably with kernels and cokernels.

5 Subcategories and quotient constructions

5.1 Serre subcategories

A Serre subcategory is a full subcategory closed under taking subobjects, quotients, and extensions. In other words, if a short exact sequence has terms inside the subcategory, then the remaining term must also lie inside it. Serre subcategories are the natural “ideal-like” structures in Abelian categories used to define quotient categories while preserving the Abelian framework.

5.2 Quotient categories and their Abelian structure

Given a Serre subcategory \(\mathcal{S}\subseteq\mathcal{A}\), one forms a quotient category \(\mathcal{A}/\mathcal{S}\) that identifies morphisms differing by those that factor through objects of \(\mathcal{S}\). Under the standard conditions that \(\mathcal{S}\) is Serre, the quotient inherits an Abelian structure. This construction is a categorical analogue of forming a “ring modulo an ideal,” but adapted to exactness and subobjects.

5.3 Localization (high-level correspondence)

Localization in Abelian categories corresponds to inverting a class of morphisms or, equivalently, forcing certain objects to become “negligible.” While detailed mechanics depend on a chosen localization scheme, the general correspondence is that localization often relates to quotienting by a subcategory of “torsion” objects or to a systematic elimination of unwanted components.

5.4 Torsion theories (brief structural connection)

A torsion theory splits objects into torsion and torsion-free parts in an exact-structure-compatible manner. Briefly, a torsion class is closed under quotients and extensions, while a torsion-free class is closed under subobjects and extensions, and the two classes are orthogonal with respect to certain Hom-vanishing properties. Abelian categories provide a natural environment where torsion theories link to localization and quotient constructions.

6 Homological algebra inside Abelian categories

6.1 Projective objects and projective resolutions

An object \(P\) is projective if \(\operatorname{Hom}(P,-)\) is exact, equivalently if lifts exist along epimorphisms. Projective resolutions express an object \(M\) as the homology of a complex built from projective objects. These resolutions allow computation of derived functors by replacing \(M\) with a projective complex.

Even when projectives are not abundant, relative projectives and resolution techniques often play the same conceptual role.

6.2 Injective objects and injective resolutions

Dually, \(I\) is injective if \(\operatorname{Hom}(-,I)\) is exact, meaning extensions along monomorphisms can be extended. Injective resolutions construct an object via a complex of injectives whose cohomology recovers the target. Injective resolutions are especially important for defining right derived functors such as Ext in many standard theories.

6.3 Ext and Tor (categorical perspective)

Ext groups measure equivalence classes of extensions and, more generally, higher derived extension phenomena. In categorical terms, Ext arises as the derived functors of \(\operatorname{Hom}\) with respect to one variable, computed using injective or projective resolutions. Tor groups measure the derived failure of tensoring to preserve exactness, typically computed via projective resolutions in module-like contexts.

The categorical perspective emphasizes that Ext and Tor are not ad hoc: they are systematically produced from exactness defects using resolutions.

6.4 Derived functors (foundational overview)

Derived functors formalize the idea that a left or right exact functor can be extended to a full sequence of higher invariants. One replaces inputs with suitable resolutions (projective for left derived, injective for right derived) and then computes homology/cohomology of the resulting complexes. Abelian categories provide the exactness setting needed to guarantee that these constructions are well-defined up to canonical isomorphism.

7 Examples and canonical sources

7.1 Modules over a ring

For a ring \(R\), the category of left \(R\)-modules is Abelian. Kernels, cokernels, finite biproducts, and exactness correspond directly to the familiar algebraic constructions. Many definitions in homological algebra are first understood in this example and then generalized by abstract category theory.

7.2 Comodules and representations (general viewpoint)

Categories of comodules over a coalgebra often behave Abelianly under suitable hypotheses, paralleling module categories. Representation categories of algebraic structures can also form Abelian categories when morphisms are linear maps compatible with the action structure. The guiding principle is that the module-like axioms extend to these settings so kernels and cokernels can be formed pointwise or functorially.

7.3 Sheaves of modules and their Abelian behavior

For a ringed space or scheme, the category of sheaves of modules over a structure sheaf is Abelian. Kernels and cokernels exist and are computed sectionwise with sheafification to maintain the sheaf property. Exact sequences of sheaves encode gluing information and are a core component of sheaf cohomology.

7.4 Chain complexes and homotopy-related constructions

While the category of chain complexes is typically additive and admits kernels and cokernels componentwise, the passage to derived or homotopy categories changes the categorical context by identifying homotopic maps or quasi-isomorphisms. Nonetheless, chain-complex categories provide an important arena where Abelian structures guide the formation of homology, cohomology, and derived functors.

8 Structural results and equivalences

8.1 Equivalence of Abelian categories (basic criteria)

Two Abelian categories may be considered “the same” up to equivalence of categories that preserves the relevant structures. Basic criteria involve the existence of a functor that is fully faithful and essentially surjective, together with control over how exact sequences are transported. Exact equivalences preserve kernels, cokernels, and the notion of exactness, so homological invariants correspond across the equivalence.

8.2 AB5 categories and Grothendieck Abelian categories

An AB5 category is an Abelian category where filtered colimits are exact (more precisely, colimits over filtered diagrams preserve exactness). Grothendieck Abelian categories are AB5 categories with additional axioms, such as the existence of a generator and often local presentability features. These conditions guarantee strong existence and continuity properties for constructions like direct limits and injective resolutions.

8.3 Noetherian and Artinian Abelian categories

An Abelian category is Noetherian if it satisfies a chain condition on subobjects resembling the classical “ACC on submodules,” ensuring that increasing sequences of subobjects stabilize. Artinian categories satisfy a dual “DCC on subobjects.” These finiteness conditions produce decomposition and structure theorems, including tendencies toward finite length objects and controlled behavior of filtrations.

8.4 Stabilization of conditions under equivalence

Many finiteness and structural properties are invariant under equivalence of Abelian categories, particularly those defined in terms of exact sequences and subobject behavior. When an equivalence is exact (or when it preserves the relevant notions of subobjects and exactness), the Noetherian or Artinian character carries over, as do AB5-type filtered colimit exactness properties.

9 Limits, colimits, and categorical completeness

9.1 Existence of finite limits and colimits

Abelian categories are complete with respect to many finite constructions: they have finite limits and finite colimits, including products, coproducts, pullbacks, and pushouts. The existence and compatibility of these operations with kernels and cokernels are essential for diagrammatic arguments and for defining exactness in terms of morphisms.

9.2 Behavior of products and coproducts

Products and coproducts exist and are related through biproducts. For finite collections, the universal properties align so that products and coproducts coincide up to canonical isomorphism, yielding a coherent direct sum picture. This simplifies the handling of exact sequences involving sums and intersections of subobjects.

9.3 Interaction with exact sequences

Exactness interacts with limits and colimits in structured ways: for instance, kernels naturally arise as pullbacks of maps to zero, and cokernels arise as pushouts from maps out of zero. Consequently, exactness can be recognized through commutative squares with suitable universal properties, making it possible to verify exactness via diagram chasing.

9.4 Pullbacks, pushouts, and exactness (schematic)

Pullbacks and pushouts encode how kernels and cokernels behave under base change–like operations. While a full statement depends on which functors preserve which colimits or limits, the schematic principle is that exactness can be tracked by embedding maps into pullback/pushout diagrams and then applying the universal characterizations of kernels and cokernels.

10 Relationship to other categorical frameworks

10.1 Exact categories vs. Abelian categories

Exact categories generalize the notion of “exactness” by designating a class of short exact sequences satisfying axioms similar to those in Abelian categories. Abelian categories provide a canonical example: their exact sequences are those arising from kernels and cokernels. Exact categories allow working in contexts where not all kernels or cokernels exist, or where exactness is restricted.

10.2 Triangulated categories (connection via derived categories)

Triangulated categories abstract the structure present in derived categories, where short exact sequences and complexes yield distinguished triangles. Abelian categories supply the underlying exactness data needed to form derived categories; from there, one obtains triangulated structures used to study homological phenomena more flexibly.

10.3 T-structures (brief contextual bridge)

T-structures on triangulated categories organize objects into an abelian-like hierarchy, often producing hearts that are Abelian categories. The bridge is that the heart of a t-structure behaves like the original Abelian category from which the derived setting began, allowing one to interpret cohomological degrees and exact sequences in the triangulated language.

10.4 Derived functors and categorical extensions

Derived functors and extension groups can be viewed as mechanisms that turn exactness into higher-order invariants. Abelian categories provide the foundational exact framework; derived categories and triangulated structures then package these invariants in a form suited for global computations and conceptual unification.

11 Diagrammatic computations and standard arguments

11.1 Snake lemma (categorical formulation)

The snake lemma is a standard result relating kernels and cokernels in a commutative diagram with exact rows. It produces a long exact sequence connecting the kernel of one map, kernels of others, and cokernels in a controlled manner. In Abelian categories, the lemma follows from the existence and universal properties of kernels and cokernels and is a central tool for diagram chasing.

11.2 3×3 lemma (usage patterns)

The 3×3 lemma concerns a commutative diagram with exact rows and columns arranged in a \(3\times 3\) grid. Under suitable exactness assumptions on some rows and columns, one can deduce exactness on the remaining ones. This lemma is widely used to streamline computations in homological algebra, particularly when dealing with extensions and iterated kernels/cokernels.

11.3 Five lemma (common applications)

The five lemma provides criteria for when a morphism between two exact sequences is an isomorphism in the middle terms, based on isomorphisms at the ends and appropriate injectivity/surjectivity conditions. It relies on how kernels and cokernels fit into exact sequences, enabling conclusions about diagram morphisms without constructing explicit inverses.

11.4 Diagram chasing techniques

Diagram chasing refers to systematic use of commutativity, universal properties, and exactness definitions to track elements or morphisms through commutative squares. In Abelian categories, kernels and cokernels supply the necessary “landmarks” so that diagram arguments can be translated into statements about existence and factorization of morphisms.

While the mechanics vary by context, the underlying idea is consistent: exactness turns algebraic constraints into structural relations among arrows in diagrams.

12 Computation toolkit and practical workflow

12.1 Checking exactness by kernels/cokernels

A practical approach to exactness in Abelian categories is to verify equality of kernels and images using the kernel/cokernel definitions. One typically checks that compositions vanish, then confirms the relevant universal properties characterize the maps involved. This method avoids reliance on external algebraic descriptions and works uniformly across examples.

12.2 Constructing kernels and cokernels explicitly (examples)

For concrete categories like modules or sheaves, kernels and cokernels can be computed explicitly (e.g., as submodules or quotient modules, or via sectionwise constructions followed by sheafification). In abstract settings, one builds them from universal properties: define candidates, show the compositions are zero, and verify the required factorization properties for all competing morphisms.

12.3 Detecting split maps and direct summands

Split monomorphisms and split epimorphisms correspond to direct summands. To detect splitting, one looks for retractions or sections compatible with the given morphism, or equivalently for the existence of an idempotent endomorphism whose image defines a summand. Once a summand is identified, exact sequences involving it often simplify to direct sum decompositions.

12.4 Using functoriality to transport exact sequences

Another workflow step is to apply functors that preserve enough exactness to transfer results. If a functor is exact, it sends short exact sequences to short exact sequences. If it is only left or right exact, one expects missing exactness at one end and then uses derived or long exact sequence constructions to recover the missing information. This “transport-and-correct” strategy is fundamental for computing Ext and Tor and for analyzing cohomological functors.