Abelian category and Grothendieck axioms AB1-AB5

Intuition Beginner

Homological algebra needs a place where exact sequences make sense. Modules over a ring are the basic example: they have maps, kernels, quotients, images, and exact sequences.

An abelian category is an abstract setting that behaves enough like modules for homological algebra to work. It may contain modules, sheaves, or other objects, but the same exact-sequence tools apply.

Grothendieck categories are abelian categories with extra infinite-sum and limit behavior. They are large enough for sheaf theory and flexible enough to contain enough injective objects.

This matters because sheaf cohomology and derived functors are built from injective resolutions, and injective resolutions require enough injectives.

Visual Beginner

The same kernel, quotient, and exactness language works across many mathematical settings.

Worked example Beginner

The category of abelian groups is abelian. A group homomorphism has a kernel, an image, and a quotient by the image.

The category of sheaves of abelian groups on a space is also abelian. Kernels and quotients are made sheaf by sheaf, with sheafification when needed.

What this tells us: abelian categories let one prove exact-sequence theorems once and use them in both algebra and geometry.

Check your understanding Beginner

Formal definition Intermediate+

A category $\mathcal{A}$ is preadditive if each morphism set $\mathrm{Hom}(X,Y)$ is an abelian group and composition is bilinear. It is additive if it is preadditive, has a zero object, and has finite biproducts.

A preabelian category is an additive category in which every morphism has a kernel and a cokernel. For a morphism $$ f\to Y, $$ one defines $$ \operatorname{coim}(f)=\operatorname{coker}(\ker f), \qquad \operatorname{im}(f)=\ker(\operatorname{coker} f). $$ An abelian category is a preabelian category in which the natural map $$ \operatorname{coim}(f)\to \operatorname{im}(f) $$ is an isomorphism for every $f$.

Grothendieck's AB axioms organize the exactness behavior of larger abelian categories:

• AB1. The category is abelian: kernels and cokernels exist and image equals coimage.
• AB2. Finite biproduct/additive exactness behaves as in module categories; in many conventions this is absorbed into the abelian-category definition.
• AB3. Small coproducts exist.
• AB4. Small coproducts are exact.
• AB5. Filtered colimits are exact.

A Grothendieck category is an abelian category satisfying AB5 and having a generator. Typical examples are ${\mathrm{Mod}}_R$, sheaves of abelian groups on a space, and many categories of quasi-coherent sheaves.

Counterexamples to common slips

• Additive does not imply abelian; kernels and cokernels may be missing.
• Preabelian does not imply abelian; image and coimage may fail to agree.
• Having coproducts does not imply that coproducts or filtered colimits are exact.

Key theorem with proof Intermediate+

Theorem (Grothendieck enough-injectives theorem). Every Grothendieck category has enough injective objects. That is, for every object $X$ there is a monomorphism $$ X\hookrightarrow I $$ with $I$ injective.

Proof. The full proof is Grothendieck's Tohoku theorem and uses the generator plus AB5 in an essential way. The idea is as follows.

The generator lets one test objects and build large objects from a controlled family of maps out of the generator. AB3 supplies the needed coproducts, while AB5 ensures that filtered colimits preserve exactness. These two infinite-exactness properties let one run a transfinite construction that keeps extending an object to solve all lifting problems against monomorphisms.

The construction is analogous in spirit to embedding a module into an injective module, but it is carried out internally to the abelian category rather than using elements of a module. The generator replaces elements, and exact filtered colimits keep the extension process exact at limit stages.

The resulting object $I$ satisfies the injective lifting property, and the original object embeds into it. Therefore the category has enough injectives. $\square$

Bridge. This theorem is the existence engine behind derived-functor cohomology. Once enough injectives exist, one can build injective resolutions and define right derived functors such as sheaf cohomology.

Exercises Intermediate+

Advanced results Master

Abelian categories are the natural habitat of exact sequences. The definition is designed so that kernels, cokernels, monomorphisms, epimorphisms, images, coimages, and short exact sequences behave as they do for modules.

This abstraction is not decorative. Derived functors are defined in abelian categories with enough injectives or enough projectives. Sheaf cohomology is the right derived functor of global sections in the abelian category of sheaves of abelian groups, and coherent-sheaf cohomology is built by applying the same exactness language in geometric categories.

Grothendieck categories add the infinite exactness needed for large categories of sheaves and modules. AB5 is especially important because filtered colimits occur constantly in sheaf and module constructions. Exactness of filtered colimits prevents hidden derived-functor errors when passing to limits of finite-stage data.

The generator condition is a size-control condition. It says there is an object $G$ such that maps from $G$ detect nonzero morphisms. In module categories, the ring $R$ as a module over itself is a generator. For sheaf categories, generators are built from sheaves supported on opens.

Grothendieck's enough-injectives theorem is one of the foundations of modern sheaf cohomology. Before it, injective resolutions were familiar for modules. Grothendieck showed that the sheaf categories needed by geometry also support injective resolutions, allowing derived functors to be defined at the right level of generality.

The AB numbering has convention variance in the literature. Some authors fold AB1 and AB2 into the definition of abelian category, then state AB3 through AB5 as extra completeness and exactness properties. The invariant content is the same: abelian exactness first, then exactness of increasingly large colimit operations.

Synthesis. Abelian categories make exact-sequence algebra portable. Grothendieck categories make it large enough for sheaves, modules, and derived functors. The enough-injectives theorem is the bridge from exactness axioms to computable cohomology theories.

Full proof set Master

Proposition 1 (module categories are abelian). For any ring $R$, the category ${\mathrm{Mod}}_R$ is abelian.

Proof. Kernels and cokernels of $R$-linear maps exist and are the usual kernel submodule and quotient by the image. The coimage is $X/\ker f$, and the image is $\mathrm{im}f\subseteq Y$. The first isomorphism theorem identifies $X/\ker f$ with $\mathrm{im}f$. Hence image and coimage agree, so ${\mathrm{Mod}}_R$ is abelian. $\square$

Proposition 2 (exactness is categorical). In an abelian category, a pair $$ A\xrightarrow{f}B\xrightarrow{g}C $$ is exact at $B$ if and only if the image of $f$ equals the kernel of $g$ as subobjects of $B$.

Proof. The abelian-category definition supplies both $\mathrm{im}(f)$ and $\ker (g)$ as subobjects of $B$. The condition $gf=0$ gives a canonical factorization of $\mathrm{im}(f)$ through $\ker (g)$. Exactness at $B$ is precisely the statement that this factorization is an isomorphism. $\square$

Proposition 3 (Grothendieck categories support derived functors). If $\mathcal{A}$ is Grothendieck and $F:\mathcal{A}\to \mathcal{B}$ is left exact, then the right derived functors $R^{i}F$ can be defined using injective resolutions.

Proof. By Grothendieck's enough-injectives theorem, every object $X$ admits an injective resolution $$ 0\to X\to I^0\to I^1\to I^2\to\cdots. $$ Apply $F$ to the resolution and take cohomology of the resulting complex. Standard comparison theorems for injective resolutions show that the result is independent of the chosen resolution up to canonical isomorphism. These cohomology objects are the right derived functors $R^{i}F(X)$. $\square$

Connections Master

• Exact sequence and snake lemma 01.02.11. Abelian categories are the setting in which exact-sequence diagram chases generalize.

• Chain complexes 01.02.30. Cohomology objects require kernels, images, and quotients in an abelian category.

• Derived functors and Ext 04.03.06. Derived functors need enough injectives or projectives.

• Triangulated categories 04.03.10. Derived and homotopy categories replace abelian exactness with triangular exactness.

• Derived category 04.03.11. The derived category starts from complexes in an abelian category.

• Grothendieck spectral sequence 04.03.13. The spectral sequence relies on derived functors in abelian/Grothendieck settings.

Historical & philosophical context Master

Grothendieck's 1957 Tohoku paper transformed homological algebra by treating abelian categories as the proper setting for derived functors and sheaf cohomology ^{[Grothendieck]}. The enough-injectives theorem was a decisive technical result: it made injective resolutions available in the categories geometry actually needed.

Cartan-Eilenberg had already axiomatized much of homological algebra for modules and related algebraic categories. Grothendieck pushed the abstraction to sheaves and functors, making cohomology a general categorical construction.

Freyd's Abelian Categories and Mac Lane's categorical foundations clarified the axioms and examples ^{[Freyd] [Mac Lane]}. Gelfand-Manin and Weibel then present abelian categories as the launch point for modern derived categories and spectral sequences ^{[Gelfand-Manin] [Weibel]}.

Philosophically, abelian categories separate the exactness pattern from the nature of the objects. Once that separation is made, modules, sheaves, representations, and quasi-coherent sheaves can share the same homological machinery.

Bibliography Master

@article{Grothendieck1957Tohoku,
  author = {Grothendieck, Alexander},
  title = {Sur quelques points d'algebre homologique},
  journal = {Tohoku Mathematical Journal},
  volume = {9},
  pages = {119--221},
  year = {1957}
}

@book{GelfandManinAbelianCategories,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  publisher = {Springer},
  year = {1996}
}

@book{Weibel1994AbelianCategories,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{Freyd1964AbelianCategories,
  author = {Freyd, Peter},
  title = {Abelian Categories},
  publisher = {Harper and Row},
  year = {1964}
}

@book{MacLaneCategoriesAbelian,
  author = {Mac Lane, Saunders},
  title = {Categories for the Working Mathematician},
  publisher = {Springer},
  year = {1971}
}