Example 9.4.1.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. We say that $U$ is idempotent complete if it satisfies either of the following equivalent conditions:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is idempotent complete.
- $(b)$
The inner fibration $U$ is $\{ \operatorname{N}_{\bullet }( \operatorname{Idem}) \} $-cocomplete, where $\operatorname{Idem}$ is the category introduced in Construction 8.5.2.7.
The implication $(b) \Rightarrow (a)$ is immediate (see the proof of Proposition 8.5.5.2). To prove the converse, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$; in this case, the result follows from the observation that the inclusion map $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}$ preserves $\operatorname{N}_{\bullet }(\operatorname{Idem})$-indexed colimiits (Corollary 8.5.3.12).