Proposition 5.1.7.13 (028A)

Proposition 5.1.7.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be inner fibrations of simplicial sets which are equivalent to one another. Then:

$(1)$: The morphism $U$ is an isofibration if and only if $V$ is an isofibration.
$(2)$: The morphism $U$ is a cartesian fibration if and only if $V$ is a cartesian fibration.
$(3)$: The morphism $U$ is a right fibration if and only if $V$ is a right fibration.
$(4)$: The morphism $U$ is a cocartesian fibration if and only if $V$ is a cocartesian fibration.
$(5)$: The morphism $U$ is a left fibration if and only if $V$ is a left fibration.
$(6)$: The morphism $U$ is a Kan fibration if and only if $V$ is a Kan fibration.

Proof. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. We first prove $(1)$. Assume that $V$ is an isofibration; we will show that $U$ is also an isofibration. Choose a monomorphism of simplicial sets $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{Q}}$, where $\operatorname{\mathcal{Q}}$ is a contractible Kan complex (Exercise 3.1.7.11). Replacing $\operatorname{\mathcal{D}}$ by the product $\operatorname{\mathcal{D}}\times \operatorname{\mathcal{Q}}$, we can assume that $F$ is a monomorphism of simplicial sets. In this case, Lemma 5.1.7.12 guarantees that $F$ exhibits $\operatorname{\mathcal{C}}$ as a retract of $\operatorname{\mathcal{D}}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$, so that $U$ is an isofibration by virtue of Remark 4.5.5.10.

To prove $(2)$, we may assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex (Proposition 5.1.4.7). In this case, $U$ and $V$ are isofibrations (Example 4.4.1.6) and $F$ is an equivalence of $\infty $-categories (Corollary 5.1.7.8). It follows from Corollary 5.1.6.2 that $U$ is a cartesian fibration if and only if $V$ is a cartesian fibration.

To prove $(3)$, suppose that $U$ is a right fibration; we will show that $V$ is a right fibration. It follows from $(2)$ that $V$ is a cartesian fibration. It will therefore suffice to show that, for each vertex $E \in \operatorname{\mathcal{E}}$, the $\infty $-category $\{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is a Kan complex (Proposition 5.1.4.14). By virtue of Remark 5.1.7.4, the morphism $F$ induces an equivalence of $\infty $-categories $F_{E}: \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$. It will therefore suffice to show that $\{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}$ is a Kan complex (Remark 4.5.1.21), which follows from our assumption that $U$ is a right fibration.

Assertions $(4)$ and $(5)$ follow by similar arguments. Assertion $(6)$ follows by combining $(3)$ and $(5)$ (see Example 4.2.1.5). $\square$