Kerodon

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Corollary 8.1.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every degenerate $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is thin.

Proof. Let $\sigma $ be a $1$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. We will show that the left-degenerate $2$-simplex $s_0(\sigma )$ is thin; a similar argument will show that the right-degenerate $2$-simplex $s_1(\sigma )$ is thin (see Remark 8.1.4.4). Unwinding the definitions, we see that $s_0(\sigma )$ corresponds to a diagram in $\operatorname{\mathcal{C}}$ of the form

\[ \xymatrix@R =50pt@C=50pt{ X \ar [dr]_{\operatorname{id}_ X} & & X \ar [dl]^{\operatorname{id}_ X} \ar [dr]_{f} & & Y \ar [dl]^{g} \\ & X \ar [dr]_{f} & & B \ar [dl]^{ \operatorname{id}_ B } & \\ & & B. & & } \]

By virtue of Proposition 8.1.6.2, it will suffice to show that the inner region of the diagram is a pushout square in $\operatorname{\mathcal{C}}$. This follows from Proposition 7.6.3.15, since $\operatorname{id}_{B}$ and $\operatorname{id}_{X}$ are isomorphisms in $\operatorname{\mathcal{C}}$. $\square$