Remark 4.5.3.17 (Axioms for Categorical Equivalence). The collection of categorical equivalences of simplicial sets has the following properties:
- $(A)$
If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a categorical equivalence if and only if it is an equivalence of $\infty $-categories (Example 4.5.3.3).
- $(B)$
Every inner anodyne morphism of simplicial sets is a categorical equivalence (Corollary 4.5.3.14).
- $(C)$
If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ have the property that two of the morphisms $f$, $g$, and $g \circ f$ are categorical equivalences, then so is the third (Remark 4.5.3.5).
In fact, the collection of categorical equivalences is characterized by assertions $(A)$, $(B)$ and $(C)$. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Invoking Proposition 4.1.3.1 twice, we can construct a commutative diagram of simplicial sets
where $u$ and $v$ are inner anodyne and $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories. It follows from $(A)$, $(B)$ and $(C)$ that the morphism $f$ is a categorical equivalence if and only if the functor $F$ is an equivalence of $\infty $-categories.