Proposition 10.3.5.12 (04YU)

Proposition 10.3.5.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which have images, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves pullback squares. The following conditions are equivalent:

$(1)$

The functor $F$ carries quotient morphisms in $\operatorname{\mathcal{C}}$ to quotient morphisms in $\operatorname{\mathcal{D}}$.

$(2)$

For every $2$-simplex $\sigma :$

\[ \xymatrix@C =50pt@R=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$ which exhibit $Y_0$ as an image of $f$, the $2$-simplex $F(\sigma )$ exhibits $F(Y_0)$ as an image of $F(f)$ in the $\infty $-category $\operatorname{\mathcal{D}}$.

$(3)$

For every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$ of Remark 10.3.5.11 carries $\operatorname{im}(f)$ to $\operatorname{im}( F(f) )$.

Proof. The implication $(1) \Rightarrow (2)$ follows from the observation that $F$ preserves monomorphisms (Proposition 9.3.4.21), and the implication $(2) \Rightarrow (3)$ is immediate from the definitions. We will complete the proof by showing that $(3)$ implies $(1)$. Let $f: X \twoheadrightarrow Y$ be a quotient morphism in $\operatorname{\mathcal{C}}$; we wish to show that $F(f)$ is a quotient morphism in $\operatorname{\mathcal{D}}$. By virtue of Proposition 10.3.3.6, our hypothesis can be reformulated as an equality $\operatorname{im}(f) = [Y]$ in the partially ordered set $\operatorname{Sub}(Y)$, and we wish to prove an equality $\operatorname{im}( F(f) ) = [ F(Y) ]$ in the partially ordered set $\operatorname{Sub}( F(Y) )$. This is clear, since the map $\operatorname{Sub}(Y) \rightarrow \operatorname{Sub}( F(Y) )$ preserves largest elements (see Remark 10.3.5.11). $\square$