Quotients of termwise split injections, for additive categories

Question

In the stack exchange notes found in Section 10 of this file, it is claimed that the category $K(\mathcal{A})$ of complexes up to homotopy is a triangulated category, if $\mathcal{A}$ is additive. In the proof of Lemma 10.2 (and earlier as well), given a termwise split injection, they form a "quotient complex" using kernels of splitting maps. I see what to do if $\mathcal{A}$ is Karoubian, but for a general additive category $\mathcal{A}$, do we know that such kernels must exist? Is there an easy fix here? Or do we need to add some hypotheses on $\mathcal{A}$?

Answer 1

The assumption that the kernel exists is built into the definition of "termwise split injection" in these notes: Definition 9.4 says

A termwise split injection $\alpha:A^\bullet\to B^\bullet$ is a morphism of complexes such that each $A_n\to B_n$ is isomorphic to the inclusion of a direct summand.

In general, such kernels certainly need not exist. For instance, you could let $\mathcal{A}$ be the full additive subcategory of $Ab$ generated by $\mathbb{Z}$ and $\mathbb{Z}\oplus \mathbb{Q}$. Then the inclusion $\mathbb{Z}\to\mathbb{Z}\oplus\mathbb{Q}$ has a splitting, but no splitting map has a kernel.