Kernel (Category Theory)

Definition

Let $\mathcal{C}$ be a category with pullbacks and an initial object $0$. Let $f:\mathcal{C}[a,b]$ be a morphism.

The kernel of $f$ is given by $\rho :\mathcal{C}[k,a]$ such that the following diagram is a pullback:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
k \arrow[d, "\rho"] \arrow[r, "!"] & 0 \arrow[d, "!"] \\
a \arrow[r, "f"] & b
\end{tikzcd}
\end{document}