Definition 5.4.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is minimal if it satisfies the following condition, for each $n \geq 0$:
- $(\ast _ n)$
Let $\sigma $ and $\sigma '$ be $n$-simplices of $\operatorname{\mathcal{C}}$. Suppose that there exists an isomorphism $h: \sigma \rightarrow \sigma '$ in the $\infty $-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})$, and that the image of $h$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$ is an identity morphism. Then $\sigma = \sigma '$.