Kerodon

Proof of Proposition 8.1.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that the comparison map $\rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$ of Construction 8.1.7.1. Let $\operatorname{\mathcal{D}}$ be a simplicial set; we will show that composition with $\rho _{+}$ induces a bijection $\theta : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )^{\simeq } )$.

Let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the projection map, and let $W$ be the collection of all edges $e$ of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ such that $\lambda _{+}(e)$ is a degenerate edge of $\operatorname{\mathcal{D}}$. Let $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those diagrams $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{C}}$ (Notation 6.3.1.1). Using Lemmas 8.1.7.9 and 8.1.7.13, we can identify $\theta $ with the map $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}})[W^{-1}], \operatorname{\mathcal{C}})^{\simeq } )$ given by composition $\lambda _{+}$. To complete the proof, it will suffice to show that $\lambda _{+}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ with respect to $W$, in the sense of Definition 6.3.1.9. This follows from Corollary 6.3.6.4, since the morphism $\lambda _{+}$ is universally localizing (Corollary 8.1.2.4). $\square$