Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 6.3.7.2. Let $S$ be a simplicial set, let $\operatorname{Sd}(S)$ denote the subdivision of $S$ (Definition 3.3.3.1), and let $\lambda _{S}: \operatorname{Sd}(S) \rightarrow S$ denote the last vertex map (Construction 3.3.4.3). Then $\lambda _{S}$ is universally localizing.

Proof of Proposition 6.3.7.2. By virtue of Proposition 6.3.6.10, we may assume without loss of generality that the simplicial set $S$ is finite. If $S$ is empty, there is nothing to prove. We may therefore assume that $S$ has dimension $n$ for some integer $n \geq 0$. We proceed by induction on $n$ and on the number of nondegenerate $n$-simplices of $S$. Fix a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow S$. Using Proposition 1.1.3.13, we see that there is a pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ S' \ar [r] & S, } \]

where $S'$ is a simplicial set of dimension $\leq n$ having fewer nondegenerate $n$-simplices than $S$. Applying Proposition 6.3.6.11 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sd}( \operatorname{\partial \Delta }^ n) \ar [dr]^{ \lambda _{\operatorname{\partial \Delta }^ n} } \ar [rr] \ar [dd] & & \operatorname{Sd}(\Delta ^ n) \ar [dr]^{ \lambda _{\Delta ^ n}} \ar [dd] & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \operatorname{Sd}(S') \ar [dr]^{ \lambda _{S'} } \ar [rr] & & \operatorname{Sd}(S) \ar [dr]^{ \lambda _{S} } & \\ & S' \ar [rr] & & S, } \]

we are reduced to showing that the morphisms $\lambda _{S'}$, $\lambda _{\operatorname{\partial \Delta }^{n}}$, and $\lambda _{\Delta ^{n}}$ are universally localizing. In the first two cases, this follows from our inductive hypothesis. We are therefore reduced to proving Proposition 6.3.7.2 in the special case where $S = \Delta ^ n$ is a standard simplex.

Using Example 3.3.3.5, we can identify the subdivision $\operatorname{Sd}(S) = \operatorname{Sd}( \Delta ^ n )$ with the nerve of the partially ordered set $\operatorname{Chain}[n]$ of nonempty subsets $P \subseteq [n]$. We wish to show that, for every morphism of simplicial sets $\alpha : T \rightarrow S$, the projection map $\pi : T \times _{S} \operatorname{Sd}(S) \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} \operatorname{Sd}(S)$ with respect to some collection of edges. By virtue of Proposition 6.3.6.2, we can assume without loss of generality that $T = \Delta ^{m}$ is a standard simplex, so that $\alpha $ can be identified with a nondecreasing map of linearly ordered sets $[m] \rightarrow [n]$. Unwinding the definitions, we can identify $T \times _{S} \operatorname{Sd}(S)$ with the nerve of the partially ordered set $A \subseteq [m] \times \operatorname{Chain}[n]$ consisting of those pairs $(i, P)$ satisfying $\max (P) = \alpha (i)$. Under this identification, the projection map $\pi $ is induced by the morphism of partially ordered sets

\[ A \rightarrow [m] \quad \quad (i,P) \mapsto i. \]

It follows that $\pi $ is a reflective localization: it has a fully faithful right adjoint, given by the construction $i \mapsto (i, \{ 0 < 1 < \cdots < \alpha (i) \} )$. The desired result is now a consequence of Proposition 6.3.3.9. $\square$