Tag Info

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 $$R$$.