Proposition 10.3.3.11 (04XD)

Proposition 10.3.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex

10.23

\begin{equation} \begin{gathered}\label{equation:universal-property-of-image} \xymatrix@R =50pt@C=50pt{ X \ar [d]^{q} \ar [dr]^{f} & \\ Y_0 \ar [r]^-{i_0} & Y } \end{gathered} \end{equation}

which exhibits $Y_0$ as an image of $f$. Then, for any monomorphism $i_1: Y_1 \hookrightarrow Y$ of $\operatorname{\mathcal{C}}$, the following conditions are equivalent:

$(1)$

The morphism $f$ factors (up to homotopy) through $i_1$. That is, there exists a $2$-simplex

10.24

\begin{equation} \begin{gathered}\label{equation:universal-property-of-image2} \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g} \ar [dr]^{f} & Y_1 \ar [d]^{i_1} \\ & Y } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{C}}$.

$(2)$

The containment $[Y_0] \subseteq [Y_1]$ holds (where we regard the isomorphism classes $[Y_0]$ and $[Y_1]$ as elements of the partially ordered set $\operatorname{Sub}(Y)$; see Notation 4.7.5.20).

Proof. The implication $(2) \Rightarrow (1)$ follows immediately from the definitions. To prove the converse, we note that in the situation of $(1)$, we can amalgamate the diagrams (10.23) and (10.24) to obtain a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{q} \ar [r]^-{g} & Y_1 \ar [d]^{i_1} \\ Y_0 \ar [r]^-{i_0} \ar@ {-->}[ur] & Y } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $q$ is a quotient morphism and $i_1$ is a monomorphism, Lemma 10.3.3.10 guarantees that this lifting problem admits an (essentially unique) solution, which proves $(2)$. $\square$