$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 6.2.4.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. Then the pullback $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a reflective subcategory of $\operatorname{\mathcal{E}}$. Moreover, a morphism $w: X \rightarrow Y$ is a $\operatorname{\mathcal{E}}'$-local equivalence if and only if it satisfies the following pair of conditions:
- $(a)$
The morphism $w$ is $U$-cocartesian.
- $(b)$
The morphism $U(w): U(X) \rightarrow U(Y)$ is a $\operatorname{\mathcal{C}}'$-local equivalence in the $\infty $-category $\operatorname{\mathcal{C}}$.
Proof.
Let $W$ be the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ which satisfy conditions $(a)$ and $(b)$. We will show that $W$ satisfies the hypotheses of Lemma 6.2.3.11:
- $(0)$
Every morphism $w: X \rightarrow Y$ belonging to $W$ is an $\operatorname{\mathcal{E}}'$-local equivalence. Fix an object $Z \in \operatorname{\mathcal{E}}'$; we wish to show that precomposition with $w$ induces a homotopy equivalence $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$. Let us abuse notation by identifying $\theta $ with the restriction map $\{ w\} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$, so that we have a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=45pt{ \{ f\} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y,Z) \ar [r]^-{ \theta } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d] \\ \{ U(w) \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) ) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y), U(Z) ) \ar [r]^-{ \overline{\theta } } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) ). } \]
Assumption $(a)$ guarantees that this diagram is a homotopy pullback square (Proposition 5.1.2.1), and assumption $(b)$ guarantees that $\overline{\theta }$ is a homotopy equivalence of Kan complexes. Applying Corollary 3.4.1.5, we conclude that $\theta $ is also a homotopy equivalence.
- $(1)$
Every isomorphism of $\operatorname{\mathcal{E}}$ belongs to $W$: this follows from Proposition 5.1.1.9 and Remark 6.2.2.4.
- $(2_{-})$
If $v: Y \rightarrow Z$ belongs to $W$ and there is a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z, } \]
then $u$ belongs to $W$ if and only if $w$ belongs to $W$. This follows from Corollary 5.1.2.4 and Remark 6.2.2.5.
- $(3)$
For every object $X \in \operatorname{\mathcal{E}}$, there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where $Y$ is contained in the subcategory $\operatorname{\mathcal{E}}'$. By virtue of our assumption that $U$ is a cocartesian fibration, it will suffice to show that there exists a $\operatorname{\mathcal{C}}'$-local equivalence $\overline{w}: U(X) \rightarrow \overline{Y}$, where $\overline{Y}$ is contained in $\operatorname{\mathcal{C}}'$ (we can then take $w$ to be a $U$-cocartesian lift of $\overline{f}$). This follows from our assumption that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective.
$\square$