Kerodon

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Variant 8.1.7.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the projection map $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ determines a morphism of simplicial sets $\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which we display informally as a diagram

\[ X_0 \xleftarrow {f_1} X_1 \xleftarrow {f_2} X_2 \leftarrow \cdots \xleftarrow {f_ n} X_ n, \]

then $\rho _{-}(\sigma )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which is depicted informally by the diagram

\[ \xymatrix@R =20pt@C=20pt{ X_0 \ar [dr]^-{\operatorname{id}} & & X_1 \ar [dr]^{\operatorname{id}} \ar [dl]^{f_1} & & \cdots \ar [dl] \ar [dr] & & X_{n-1} \ar [dr]^{\operatorname{id}} \ar [dl]^{f_{n-1}} & & X_ n \ar [dl]^{f_ n} \\ & X_0 \ar [dr]^{\operatorname{id}} & & X_1 \ar [dl]^{f_1} \ar [dr]^{\operatorname{id}} & \cdots & X_{n-2} \ar [dl]^{f_{n-2}} \ar [dr]^{\operatorname{id}} & & X_{n-1} \ar [dl]^{f_{n-1}} & \\ & & \cdots \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{f_1} \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{f_2} & & \\ & & & X_{0} \ar [dr]^{\operatorname{id}} & & X_{1} \ar [dl]^{f_1} & & & \\ & & & & X_0. & & & & } \]

If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\rho _{-}$ restricts an equivalence $\operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}})$, where $\operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}( \operatorname{\mathcal{C}})$ is the simplicial subset $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$ where $L$ is the collection of isomorphisms in $\operatorname{\mathcal{C}}$, and $R$ is the collection of all morphisms of $\operatorname{\mathcal{C}}$.