Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.3.4.8. Let $X$ be a simplicial set. Then the last vertex map $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ is a weak homotopy equivalence.

Proof of Proposition 3.3.4.8. For each integer $n \geq 0$, let $\operatorname{sk}_{n}(X)$ denote the $n$-skeleton of the simplicial set $X$. Then the last vertex map $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ can be realized as a filtered colimit of the last vertex maps $\lambda _{ \operatorname{sk}_ n(X)}: \operatorname{Sd}( \operatorname{sk}_{n}(X) ) \rightarrow \operatorname{sk}_{n}(X)$. Since the collection of weak homotopy equivalences is closed under the formation of filtered colimits (Proposition 3.2.7.3), it will suffice to show that each of the maps $\lambda _{ \operatorname{sk}_{n}(X)}$ is a weak homotopy equivalence. We may therefore replace $X$ by $\operatorname{sk}_{n}(X)$, and thereby reduce to the case where $X$ is $n$-skeletal for some nonnegative integer $n \geq 0$. We proceed by induction on $n$. If $n=0$, then the simplicial set $X$ is discrete and $\lambda _{X}$ is an isomorphism (Example 3.3.4.6). We will therefore assume that $n > 0$.

Fix a Kan complex $Q$; we wish to show that composition with $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ induces a bijection $\pi _0( \operatorname{Fun}(X, Q) ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{Sd}(X), Q) )$. In fact, we will show that the map $\operatorname{Fun}(X,Q) \rightarrow \operatorname{Fun}( \operatorname{Sd}(X), Q)$ is a weak homotopy equivalence. Let $Y = \operatorname{sk}_{n-1}(X)$ be the $(n-1)$-skeleton of $X$, so that we have a commutative diagram

\[ \xymatrix { \operatorname{Fun}( X, Q) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}(\operatorname{Sd}(X), Q) \ar [d] \\ \operatorname{Fun}( Y, Q) \ar [r] & \operatorname{Fun}( \operatorname{Sd}(Y), Q), } \]

where the lower horizontal map is a homotopy equivalence by virtue of our inductive hypothesis (together with Corollary 3.1.6.5). It will therefore suffice to show that, for every morphism of simplicial sets $f: Y \rightarrow Q$, the induced map of fibers

\[ \theta _{f}: \{ f \} \times _{ \operatorname{Fun}(Y,Q)} \operatorname{Fun}(X,Q) \rightarrow \{ f\} \times _{ \operatorname{Fun}( \operatorname{Sd}(Y), Q) } \operatorname{Fun}( \operatorname{Sd}(X), Q) \]

is a homotopy equivalence (Proposition 3.2.7.1).

Let $S$ denote the collection of nondegenerate $n$-simplices of $X$, let $X' = \coprod _{\sigma \in S} \Delta ^{n}$ denote their coproduct, and let $Y' = \coprod _{\sigma \in S} \operatorname{\partial \Delta }^{n}$ denote the boundary of $X'$. Proposition 1.1.3.13 then supplies a pushout diagram of simplicial sets

\[ \xymatrix { \underset { \sigma \in S }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in S}{\coprod } \Delta ^{n} \ar [d] \\ Y \ar [r] & X, } \]

which we can use to identify $\theta _{f}$ with the induced map

\[ \theta '_{f}: \{ f \} \times _{ \operatorname{Fun}(Y',Q)} \operatorname{Fun}(X',Q) \rightarrow \{ f\} \times _{ \operatorname{Fun}( \operatorname{Sd}(Y'), Q) } \operatorname{Fun}( \operatorname{Sd}(X'), Q). \]

Invoking Proposition 3.2.7.1 again, we are reduced to showing that the horizontal maps appearing in the diagram

\[ \xymatrix { \operatorname{Fun}( X', Q) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}(\operatorname{Sd}(X'), Q) \ar [d] \\ \operatorname{Fun}( Y', Q) \ar [r] & \operatorname{Fun}( \operatorname{Sd}(Y'), Q) } \]

are homotopy equivalences. By virtue of Corollary 3.1.6.5, it will suffice to show that the last vertex maps $\lambda _{Y'}: \operatorname{Sd}(Y') \rightarrow Y'$ and $\lambda _{X'}: \operatorname{Sd}(X') \rightarrow X'$ are weak homotopy equivalences. In the first case, this follows from our inductive hypothesis (since $Y'$ has dimension $< n$). In the second, we can use Remark 3.1.5.19 to reduce to the problem of showing that the last vertex map $\lambda _{ \Delta ^{n} }: \operatorname{Sd}( \Delta ^{n} ) \rightarrow \Delta ^{n}$ is a weak homotopy equivalence. This is clear, since both $\operatorname{Sd}( \Delta ^{n} )$ and $\Delta ^{n}$ are contractible by virtue of Example 3.2.6.6 (they can be realized as the nerves of partially ordered sets $\operatorname{Chain}[n]$ and $[n]$, each of which has a largest element). $\square$