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Use this tag for questions about Abelian categories, which are categories that possess most of properties of categories of modules over a ring, and are easy to work with using techniques of homological algebra.

Definition

The definition of abelian category is defined so as to model all the nice properties that a category of modules over a ring has. Explicitly, a category $\mathcal{C}$ is abelian if:

  • It is preadditive, which means that for any two objects $X$ and $Y$ of $\mathcal{C}$, $\mathrm{Hom}(X,Y)$ has the structure of an abelian group. In a different language, this is the same as saying that $\mathcal{C}$ is enriched over $\text{Ab}$, the category of abelian groups.

  • It has a zero object $\mathbb{0}$, which is an object that is both initial and terminal in the category. Explicitly, for any object $X$ of $\mathcal{C}$, there is a unique morphism $\mathbb{0} \to X$, and there is a unique morphisms $X \to \mathbb{0}$.

  • The biproduct of any finite collection of objects in $\mathcal{C}$ exists. So for a collection $\{X_1,\dotsc,X_n\}$ of objects of $\mathcal{C}$, there is some object $\bigoplus_j X_j$ that is both the categorical product and coproduct of the $\{X_j\}_j$.

  • Every morphism has a kernel and a cokernel.

  • Lastly, every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism.

There are quite a few other equivalent characterizations of abelian categories too. These are typically the categories that we consider when studying homological algebra.

Examples & Conterexamples

  • For a ring $R$, the category $R\text{-Mod}$ of left $R$ modules is abelian. Each hom-set $\mathrm{Hom}(X,Y)$ of $R$-module homomorphisms inherits the structure of an abelian group from the abelian group structure on $Y$. The zero object of $R\text{-Mod}$ is the trivial module $\{0\}$ consisting of a single element. The biproduct of modules is usually called their direct sum. Any $R$-module homomorphisms admits a kernel and cokernel. And if you have a monomorphism $\varphi\colon N \hookrightarrow M$ then $\varphi$ is the kernel of the quotient map $M \to M/N$, and for an epimorphism $\psi\colon M \twoheadrightarrow L$ the map $\mathrm{Ker}\psi \to M$ will have cokernel $\psi$.

  • The category $\text{Ab}$ of abelian groups is an abelian category, since it is the module category $\mathbf{Z}\text{-Mod}$.

  • Given a quiver $Q$ and a field $\mathbf{k}$, the category $\mathrm{Rep}_{\mathbf{k}}Q$ of all $\mathbf{k}$-linear representations of $Q$ is an abelian category. This can be seen by directly showing that this category of representations is equivalent to a module category, the category of modules over the path algebra of $Q$.

Fundamental Theorems

The Freyd-Mitchell Embedding Theorem (nLab) is a precises statement of the catchphrase that abelian categories are "like" module categories.

Theorem — Every small abelian category admits a full, faithful, and exact functor to a category $R\text{-Mod}$ for some ring $R$.

Helpful Resources