<-- Go Back Last Updated: 11/01/2025

Kernel**

In an additive category \(\mathcal{A}\), the 'kernel' of an arrow \(f:X\to Y\) refers to both an object \(\mathrm{ker}(f)\in\mathcal{A}_0\) and an arrow \((\mathrm{ker}(f):\mathrm{ker}(f)\to X)\in \mathcal{A}_1\). The kernel of \(f\) is the equaliser between \(f\) and the \(0\)-map from \(X\) to \(Y\), the equaliser being a specific example of a limit: given any object \(Z\in\mathcal{A}_0\) and any arrow \((g:Z\to X)\in\mathcal{A}_1\) such that \(f\circ g=0\circ g=0\), there exists a unique arrow \(\tilde{g}:Z\to\mathrm{ker}(f)\) such that \(g=\mathrm{ker}(f)\circ \tilde{g}\) - any map which vanishes upon composition with \(f\) factors through the kernel of \(f\).