# Kerodon

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Corollary 8.1.2.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the projection map $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is universally localizing (see Definition 6.3.6.1).

Proof. Writing $\operatorname{\mathcal{C}}$ as the filtered colimit of its skeleta $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ and using Proposition 6.3.6.12, we can reduce to the case where $\operatorname{\mathcal{C}}$ has dimension $\leq n$ for some integer $n \geq 0$. We proceed by induction on $n$. If $n = 0$, the morphism $\lambda _{+}$ is an isomorphism. Let us therefore assume that $n$ is positive. Let $S$ denote the collection of nondegenerate $n$-simplices of $\operatorname{\mathcal{C}}$, so that Proposition 1.1.3.13 supplies a pushout square

$\xymatrix@R =50pt@C=50pt{ S \times \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & S \times \Delta ^ n \ar [d] \\ \operatorname{sk}_{n-1}(\operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}, }$

where the horizontal maps are monomorphisms. Combining our inductive hypothesis with Proposition 6.3.6.13, we can replace $\operatorname{\mathcal{C}}$ by $S \times \Delta ^ n$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty$-category. In this case, $\lambda _{+}$ is a cocartesian fibration (Corollary 8.1.1.14) having weakly contractible fibers (Proposition 8.1.2.1 and Corollary 4.6.7.25), and is therefore universally localizing by virtue of Example 6.3.6.2. $\square$