Kerodon

Warning 8.5.4.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ is idempotent complete, then its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be idempotent complete. For example, the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ is idempotent complete (Example 8.5.4.20), but its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ is not (see Proposition 8.5.7.15).