Example 5.4.2.2. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $q$ is an interior fibration (in the sense of Definition 5.4.2.1).
- $(2)$
The morphism $q$ is an inner fibration (in the sense of Definition 4.1.1.1).
The implication $(1) \Rightarrow (2)$ follows from the observation that every $2$-simplex of $\operatorname{\mathcal{D}}$ is thin, and the implication $(2) \Rightarrow (1)$ follows from Corollary 5.1.1.10. In particular, if either of these conditions is satisfied, then $\operatorname{\mathcal{C}}$ is an $\infty $-category.