In this short post we define abelian categories (which takes some work), which were introduced as abstractions of some core properties of categories like abelian groups and modules over a ring; so for example most of homological algebra can be carried over any abelian category, which is neat.

1. A "sum-free" definition

In this section we give a definition that is agnostic to a summation operation, but turns out to be equivalent to a certain nice class of categories enriched over $\underline{\mathbf{AbGrp}}$ (which we might show in a later post). To get to the definition, we first need to set up some categorical vocabulary.

Definition 1.

A zero object in a category is an object that is both initial and terminal.

A category with a zero object is sometimes called a pointed category. So for example the category $\underline{\mathbf{Set}}$ is not pointed, but the category $\underline{\mathbf{Grp}}$ is via the zero group.

Definition 2.

A constant morphism $f:A\to B$ is one such that for any $g,h:C\to A$ one has $fg=fh$, and a coconstant morphism $f:A\to B$ is one such that, dually, for any $g,h:B\to C$ one has $gf=hf$. A morphism is called a zero morphism if it's both constant and coconstant.

Notice that constant morphisms are like morphisms into a terminal object, for indeed if $B$ is terminal in the above definition, $fg=fh$ must hold; similarly, coconstant morphisms are like morphisms out of an initial object.

Definition 3.

We say that $\mathscr{C}$ has zero morphisms if for every $A,B\in\mathop{\rm ob}\nolimits\mathscr{C}$ there is a distinguished morphism $0_{AB}:A\to B$, such that the morphisms $0_{AB}$ are a collection of compatible zero morphisms; i.e., such that for any triple $X,Y,Z\in\mathop{\rm ob}\nolimits\mathscr{C}$ and all $f:Y\to Z, g:X\to Y$ the diagram \begin{align*}

\begin{xy} \xymatrix{ X\ar[r]^{0_{XY}}\ar[d]_g \ar[dr]^{0_{XZ}}& Y\ar[d]^f \\ Y\ar[r]^{0_{YZ}} & Z } \end{xy}
\end{align*} commutes.

These are rather horrible definitions, so here are some things that make it easier to think about zero morphisms; in particular, things make more sense in a category with a zero object.

Lemma 1.

If $\mathscr{C}$ has zero morphisms as above, then

1. The morphisms $0_{AB}$ are zero morphisms;
2. The collection of morphisms $0_{AB}$ is unique.

Proof 1.

1. If $g,h:C\to A$, from $C \xrightarrow{ \ g \ } A \xrightarrow{ \ 0_{AB} \ } B$ we conclude that $0_{CB} = 0_{AB}g$ and similarly $0_{CB}=0_{AB}h$, so $0_{AB}$ is constant, and coconstant analogously (or because having zero morphisms is self-dual).
2. This is sort of like proving that identities are unique in a group. Suppose $0_{AB}$ and $\widehat{0}_{AB}$ are two collections of morphisms as in the definition of $\mathscr{C}$ having zero morphisms. So take any two $A,B\in\mathop{\rm ob}\nolimits\mathscr{C}$, and an arbitrary additional $C\in\mathscr{C}$, and observe that the two commutative squares we get by using either collection and taking zero maps from the other collection for $f$ and $g$ tell us that \begin{align*} 0_{AB} = 0_{CB}\widehat{0}_{AC} = \widehat{0}_{AB} \end{align*}

Lemma 2.

If $\mathscr{C}$ has a zero object $Z$, for every $A,B\in\mathop{\rm ob}\nolimits\mathscr{C}$ there is a zero morphism $0_{AB}:A\to B$ which is the unique morphism that factors through $Z$ as $0_{AB}:A\to Z\to B$ (whereas in a general category there may be multiple zero morphisms between two objects).

Proof 2.

Denote the unique morphism $X\to Z$ by $0_{XZ}$, and $Z\to X$ by $0_{ZX}$. Then the morphism $0_{ZB}0_{AZ}$ is a zero morphism $A\to B$: indeed, if we have $f,g:C\to A$, then \begin{align*} 0_{ZB}0_{AZ}f = 0_{ZB}0_{CZ} = 0_{ZB}0_{AZ}g \end{align*} and similarly for the dual statement.

Conversely, suppose we have two zero morphisms $f,g:A\to B$. Then consider the two maps $\mathop{\rm id}\nolimits_A:A\to A$ and $0_{ZA}0_{AZ}:A\to A$; we get \begin{align*} f = f\mathop{\rm id}\nolimits_A = f0_{ZA}0_{AZ} = 0_{ZB}0_{AZ} = g0_{ZA}0_{AZ} = g\mathop{\rm id}\nolimits_A = g \end{align*} and hence there is at most one zero morphism $A\to B$. Joining the two arguments, we're done.

The notion of zero morphisms allows us to give a categorical notion of kernels and cokernels by equalizing/coequalizing with a zero morphism, as one would expect:

Definition 4.

For $\mathscr{C}$ with zero morphisms and $f:A\to B$, a kernel of $f$ is an equalizer of $f$ and $0_{XY}$, and a cokernel of $f$ is a coequalizer of $f$ and $0_{XY}$.

The last bit of niceness we need is for epimorphisms and monomorphisms to behave as we're used to over abelian groups, modules, etc., that is, to come from kernels and cokernels. This is captured by the property of normality:

Definition 5.

A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism.

With all that, we can finally say what we want to say:

Definition 6.

An abelian category is a category that:

• Has a zero object;
• Has all binary products and coproducts;
• Has all kernels and cokernels;
• all monomorphisms are normal and all epimorphisms are conormal