# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Lemma 8.1.7.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories and let $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ be a functor, corresponding to a morphism of simplicial sets $f: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which satisfy the left two-out-of-three property. Then $f$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if $F$ satisfies the following pair of conditions:

$(1)$

For each object $X$ in $\operatorname{\mathcal{D}}$, the morphism $F$ carries every morphism of $\{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $L$.

$(2)$

For each vertex $Y$ in $\operatorname{\mathcal{D}}$, the morphism $F$ carries every morphism of $\operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ Y\}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $R$.

Proof. For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, let $\operatorname{id}_{X} \xrightarrow { e_{L} } e \xleftarrow { e_{R} } \operatorname{id}_{Y}$ be the tautological cospan in $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. By virtue of Remark 8.1.6.3, the morphism $f$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if it satisfies the following pair of conditions:

$(1')$

For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, the functor $F$ carries $e_{L}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $L$.

$(2')$

For every morphism $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, the functor $F$ carries $e_{R}$ to a morphism of $\operatorname{\mathcal{C}}$ which belongs to $R$.

The implications $(1) \Rightarrow (1')$ and $(2) \Rightarrow (2')$ are immediate (if $e: X \rightarrow Y$ is any morphism of $\operatorname{\mathcal{D}}$, then $e_{L}$ is contained to the fiber $\{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$, and $e_{R}$ is contained in $\operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ Y\}$ ). We will complete the proof by showing that $(1')$ implies $(1)$; a similar argument shows that $(2')$ implies $(2)$.

Assume that condition $(1')$ is satisfied, let $X$ be an object of $\operatorname{\mathcal{D}}$, and let $u: X \rightarrow Y$ and $v: X \rightarrow Z$ be morphisms of $\operatorname{\mathcal{D}}$. Suppose we are given a morphism $g: u \rightarrow v$ in the $\infty$-category $\operatorname{\mathcal{E}}= \{ X\} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}})$; we wish to show that $F(g)$ belongs to $L$. Since $\operatorname{id}_{X}$ is initial when viewed as an object of $\operatorname{\mathcal{E}}$ (Proposition 8.1.2.1), there is a $2$-simplex of $\operatorname{\mathcal{E}}$ whose boundary is indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & \operatorname{id}_{X} \ar [dl]^{ u_{L} } \ar [dr]_{ v_{L} } & \\ u \ar [rr]^{g} & & v. }$

Assumption $(1')$ guarantees that $F(u_{L} )$ and $F(v_ L)$ belong to $L$. Since $L$ satisfies the left two-out-of-three property, it follows that $F(g)$ also belongs to $L$. $\square$