[ct/7] Abelian Categories

Kernels and cokernels

A null object in category \(C\) is an object that is both initial and final. If \(C\) has a null object, then to any \(a, b \in C\), there exist unique arrows \(a \rightarrow z\), and \(z \rightarrow b\). A null object is unique upto isomorphism.

Let \(C\) have a null object. The kernel of arrow \(f : a \rightarrow b\) is defined to be an equalizer of arrows \(f, 0 : a \rightrightarrows b\). Put more directly, \(k : s \rightarrow a\) is the kernel of \(f : a \rightarrow b\) when \(fk = 0\), and every arrow \(h\) such that \(fh = 0\) factors uniquely through \(k\):

\[ \begin{xy} \xymatrix{ s\ar[dr]_k\ar[drr]^0 & & \\ & a\ar[r]^<<{f} & b \\ c\ar[ur]^h\ar[urr]_0\ar@{.>}[uu]|{\exists!} & & } \end{xy} \]

Thus, any category with equalizers, or more generally finite limits, and with zero has kernels for all arrows, and the arrow \(k : s \rightarrow a\) is unique upto isomorphism of \(s\). Like all equalizers, the kernel is necessarily a [monic](/ct/1#monics-epis-and-zeros). It is convenient, therefore, to think of the kernel as a subobject of \(a\), or the equivalence class of monics \(s \rightarrow a\).

For example, in \(\textbf{Grp}\) with just one element (the identity element \(I\)) is a null object, and for any two groups with zero morphism, \(G \rightarrow H\) is the unique morphism which sends all of \(G\) to the identity in \(H\). The kernel of an arbirary morphism \(f : G \rightarrow H\) is the insertion \(N \rightarrow G\) of the usual kernel \(N =\) all \(x\) in \(G\) with \(fx = 1\). Note that \(N\) is the normal subgroup, and that in \(\textbf{Grp}\), all kernels are monic, but there are monics which are not kernels.

In the category \(\textbf{Set}_*\) of pointed sets, the one-point set is a null object, and zero map \(P \rightarrow Q\) is the function taking all of \(P\) to the base \(*_Q\) in \(Q\). For any morphism \(f : P \rightarrow Q\) of pointed sets, its kernel \(S \rightarrow P\) is the insertion of the subset \(S\) of those \(S \in P\) with \(fx = *_Q\), where base point of \(S\) is identical with base point \(P\). The same discussion gives kernels in \(\textbf{Top}_*\). In \(\textbf{Grp}\), an epimorphism is determined by its kernel, but this is not the case for \(\textbf{Set}_*\) or \(\textbf{Top}_*\).

In an \(\text{Ab}\)-category \(A\), all equalizers are kernels. Indeed, in such a category, each hom-set \(A(b, c)\) is an abelian group. Given parallel arrows \(f, g : b \rightarrow c\) and arrow \(h : a \rightarrow b\), such that \(fh = gh\) iff \((f - g)h = 0\). Therefore, the universal \(h\) can either be described as the equalizer of \(f, g\), or kernel of \(f - g\). The dual notion gives rise to cokernel, as [already discussed](/ct/2#coproducts-and-colimits).