Remark 9.2.8.12. Let $X$ be a Kan complex. The following conditions are equivalent:
- $(1)$
The Kan complex $X$ is finitely dominated in the sense of Definition 9.2.7.7: that is, it is a retract of an essentially finite Kan complex $Y$ (in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$).
- $(2)$
The Kan complex $X$ is finitely dominated when regarded as an $\infty $-category, in the sense of Definition 9.2.8.11: that is, it is a retract of an essentially finite $\infty $-category $\operatorname{\mathcal{C}}$ (in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$).
The implication $(1) \Rightarrow (2)$ is immediate. For the converse, assume that $X$ is a retract of an essentially finite $\infty $-category $\operatorname{\mathcal{C}}$. Then $X$ is also a retract of the Kan complex $\operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$, which is also essentially finite (Remark 9.2.8.5).