In the stack exchange notes found in Section 10 of [this file](http://stacks.math.columbia.edu/download/derived.pdf), 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 $\mathcal{A}$?



Quotients of termwise split injections, for additive categories
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.
> 
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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.


