Definition 4.7.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. We say that $\operatorname{\mathcal{C}}$ is minimal in dimension $n$ if it satisfies the following condition:
- $(\ast _ n)$
Let $\sigma _0, \sigma _1: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be $n$-simplices of $\operatorname{\mathcal{C}}$. If $\sigma _0$ is isomorphic to $\sigma _1$ relative to $\operatorname{\partial \Delta }^ n$, then $\sigma _0 = \sigma _1$.
We say that $\operatorname{\mathcal{C}}$ is minimal if it is minimal in dimension $n$ for every integer $n$.