PlanetMath: kernel is an inverse limit

Proof. Recalling the definition of a kernel in an abelian category, we can see that $i\colon X\to A$ is a kernel of

if and only if the following diagrams commute:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X \ar[dl]_i \ar[dr]^{f\circ i}&\ A \ar[rr]^f & & B } } \end{xy}$

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X \ar[dl]_i \ar[dr]^{0\circ i}&\ A \ar[rr]^0 & & B } } \end{xy}$

and $f\circ i = 0\circ i$ .

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \diamondsuit \ar@<0.5ex>[r]^\dagger \ar@<-0.5ex>[r]_\ddagger & \heartsuit } } \end{xy}$

and define a functor $G\colon I\to C$ by $G(\diamondsuit)=A$ , $G(\heartsuit)=B$ , $G(\dagger) = f$ , and $G(\ddagger) = 0$ .

Suppose that ${\mathrm{\varprojlim}}G$ exists and is equal to . Then there are maps $\pi_\diamondsuit\colon Y\to A$ and $\pi_\heartsuit\colon Y\to A$ that make the following diagrams commute:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & Y \ar[dl]_{\pi_\diamondsuit} \ar[dr]^{\pi_\heartsuit}&\ A \ar[rr]^{G(\dagger)=f} & & B } } \end{xy}$

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & Y \ar[dl]_{\pi_\diamondsuit} \ar[dr]^{\pi_\heartsuit}&\ A \ar[rr]^{G(\ddagger)=0} & & B } } \end{xy}$

This is exactly the universal condition for a kernel in an abelian category. $\qedsymbol$

This result can be extremely useful in proving exactness results: one shows that finite inverse and direct limits exist and are exact in a particular category, and one immediately obtains the fact that sums, products, kernels and cokernels are all exact.