Lemma 45.4.2. The category $M_ k$ is Karoubian.

**Proof.**
Let $M = (X, p, m)$ be a motive and let $a \in \mathop{\mathrm{Mor}}\nolimits (M, M)$ be a projector. Then $a = a \circ a$ both in $\mathop{\mathrm{Mor}}\nolimits (M, M)$ as well as in $\text{Corr}^0(X, X)$. Set $N = (X, a, m)$. Since we have $a = p \circ a \circ a$ in $\text{Corr}^0(X, X)$ we see that $a : N \to M$ is a morphism of $M_ k$. Next, suppose that $b : (Y, q, n) \to M$ is a morphism such that $(1 - a) \circ b = 0$. Then $b = a \circ b$ as well as $b = b \circ q$. Hence $b$ is a morphism $b : (Y, q, n) \to N$. Thus we see that the projector $1 - a$ has a kernel, namely $N$ and we find that $M_ k$ is Karoubian, see Homology, Definition 12.4.1.
$\square$

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