The following question is taken from
"Representation theory of the Virasoro Algebra" by Iohara and Koga, and $\textit{Theory of Categories}$ by Nicolae Popescu and Liliana Popescu
$\color{Green}{Background:}$
[From Virasoro]
$\textbf{(1):}$ A pair $(K,i)$ is called the $\textit{kernel}$ of $f$ if it satisfies
i. $f\circ i=0$ and
ii. for any morphism such that $f\circ j=0$ there exists a unique morphism $k$ such that $j=i\circ k.$
In such case, the object $K$ is often denoted by $\text{ker }f$
$\textbf{(2):}$ A pair $(C,p)$ is called the $\textit{cokernel}$ of $f$ if it satisfies
i. $p\circ f=0$ and
ii. for any morphism such that $q\circ f=0$ there exists a unique morphism $r$ such that $q=r\circ p.$
In such case, the object $C$ is often denoted by $\text{Coker }f$
[From Nicolae Popescu and Liliana Popescu]
$\textbf{(3):}$ Definition of Kernel Pair
$\begin{array}{ccccccccc} Z & \xrightarrow{u} & X\\
\small {v}\big\downarrow & & \big\downarrow\small {f} & \\
X & \xrightarrow{f} & Y
\end{array}$
Let $f:X\rightarrow Y$ be a morphism. An ordered pair $(u,v):Z\rightarrow X$ of morphisms is called a $\textit{kernel pair}$ of $f$ if $f\circ u=f\circ v$, and furthermore if for each ordered pair $(u',v'):Z'\rightarrow X$ of morphisms, such that $f\circ u'=f\circ v',$ there exists a unique morphism $g:Z'\rightarrow Z$ with $u\circ g=u'$ and $v\circ g=v'.$
$\textbf{(4):}$ Definition of Cokernel Pair
$\begin{array}{ccccccccc} Z & \xleftarrow{s} & X\\
\small {r}\big\uparrow & & \big\uparrow\small {p} & \\
X & \xleftarrow{p} & Y
\end{array}$
Let $p:Y\rightarrow X$ be a morphism. An ordered pair $(r,s):X\rightarrow Z$ of morphisms is called a $\textit{cokernel pair}$ of $p$ if $r\circ p=s\circ p$, and furthermore if for each ordered pair $(r',s'):X\rightarrow Z'$ of morphisms, such that $r'\circ p=s'\circ p,$ there exists a unique morphism $w:Z\rightarrow Z'$ with $w\circ r=r'$ and $w\circ s=s'.$
$\color{Red}{Questions:}$
What I would like to know is, given the definitions of kernel, cokernel, why are the definition of kernel and cokernel pairs needed. Everytime this this comes up in category theory text, there are no examples illustrating the differences between kernel and kernel pair or cokernel and cokernel pair. Can someone explain to me the motivation behind kernel pairs and cokernel pairs and illustrate with example. Thank you in advance
Best Answer
I can give you some intuition about the difference between the kernel pair and the kernel of a map. Let us look at the category of groups for example. Given a morphism of groups $f:G\to H$ the kernel of that map will be the subgroup $\{g\in G|f(g) = 1_H$} of $G$. In contrast, the kernel pair of $f$ will be a subgroup of $G\times G$. It consists of those pairs $(g_1,g_2)$ for which $f(g_1) = f(g_2)$. The overall kernel pair diagram looks as follows.
I have used notation that describes the underlying sets and functions, but the group structure on the kernel pair is the one induced by the entrywise group structure on $G\times G$. What you see is that the kernel pair of $f$ collects information about all pairs of elements of $G$ which get identified by $f$. The kernel in contrast only collects information about the elements of $G$ which get identified with $1_H$. People use the kernel when they do group theory, because the information in the kernel pair can be recovered from the kernel. We have that $f(a) = f(b)$ for any two elements $a,b\in G$ if and only if $f(ab^{-1}) = 1_H$. One can, in a similar way recover the kernel pair of map $f$ from its kernel when $f$ is for example a morphism of $R$-modules.
In situations where there isn't a good notion of a kernel which allows you to recover the kernel pair (all non-algebraic situations, categories without a zero object) one can still use the kernel pair to measure how much a morphism $f$ fails to be a monomorphism. You can show as an exercise that in a category with finite limits a morphism $f:X\to Y$ is monic if and only if the following is a pullback (i.e. if and only if the kernel pair is the diagonal of $X\times X$)