Notation 3.3.4.1. Let $Q$ be a partially ordered set. Every finite, nonempty, linearly ordered subset $S \subseteq Q$ has a largest element, which we will denote by $\mathrm{Max}(S)$. The construction $S \mapsto \mathrm{Max}(S)$ determines a nondecreasing function $\mathrm{Max}: \operatorname{Chain}[Q] \rightarrow Q$, where $\operatorname{Chain}[Q]$ is defined as in Notation 3.3.2.1.
3.3.4 The Last Vertex Map
Let $X$ be a simplicial set and let $\operatorname{Sd}(X)$ denote its subdivision (Notation 3.3.3.2). If $X$ is braced, then Proposition 3.3.3.6 supplies a canonical homeomorphism of topological spaces $| \operatorname{Sd}(X) | \simeq |X|$. Beware that $X$ and $\operatorname{Sd}(X)$ need not be isomorphic as simplicial sets: for example, the standard simplex $X = \Delta ^{n}$ has $n+1$ vertices, while subdivision $\operatorname{Sd}(\Delta ^{n} )$ has $2^{n+1} - 1$ vertices. Nevertheless, we will prove in this section that $X$ and $\operatorname{Sd}(X)$ are weakly homotopy equivalent. More precisely, for every simplicial set $X$ there is a canonical weak homotopy equivalence $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$, which we refer to as the last vertex map (Construction 3.3.4.3).
Remark 3.3.4.2. Let $f: P \rightarrow Q$ be a nondecreasing function between partially ordered sets. Then the diagram of partially ordered sets is commutative.
Construction 3.3.4.3. Let $X$ be a simplicial set. For every $n$-simplex $\sigma : \Delta ^ n \rightarrow X$, we let $\rho _{X}(\sigma )$ denote the composite map which we regard as an $n$-simplex of the simplicial set $\operatorname{Ex}(X)$ of Construction 3.3.2.5. It follows from Remark 3.3.4.2 that the construction $\sigma \mapsto \rho _{X}(\sigma )$ determines a map of simplicial sets $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$. Let $u: X \rightarrow \operatorname{Ex}( \operatorname{Sd}(X) )$ be a map of simplicial sets which exhibits $\operatorname{Sd}(X)$ as a subdivision of $X$ (Definition 3.3.3.1). Then there is a unique map of simplicial sets $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ for which the composition $X \rightarrow \operatorname{Ex}( \operatorname{Sd}(X) ) \xrightarrow { \operatorname{Ex}( \lambda _ X ) } \operatorname{Ex}(X)$ is equal to $\rho _{X}$. We will refer to $\lambda _{X}$ as the last vertex map of $X$.
Remark 3.3.4.4 (Functoriality). The morphisms $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ and $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ depend functorially on the simplicial set $X$. That is, for every map of simplicial sets $f: X \rightarrow Y$, the diagrams are commutative. We may therefore regard the constructions $X \mapsto \rho _{X}$ and $X \mapsto \lambda _{X}$ as natural transformations of functors
Example 3.3.4.5. Let $Q$ be a partially ordered set, so that we can identify the subdivision of $\operatorname{N}_{\bullet }(Q)$ with the nerve of the partially ordered set $\operatorname{Chain}[Q]$ (Example 3.3.3.17). Under this identification, the last vertex map $\lambda _{ \operatorname{N}_{\bullet }(Q)}$ corresponds to the morphism $\operatorname{N}_{\bullet }( \operatorname{Chain}[Q] ) \rightarrow \operatorname{N}_{\bullet }( Q)$ induced by $\mathrm{Max}: \operatorname{Chain}[Q] \rightarrow Q$.
Example 3.3.4.6. Let $X$ be a discrete simplicial set (Definition 1.1.5.10). Then the maps are isomorphisms.
Example 3.3.4.7. Let $X$ be a braced simplicial set, so that the subdivision $\operatorname{Sd}(X)$ can be identified with the nerve of the category $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ of nondegenerate simplices of $X$ (Proposition 3.3.3.16). Under this identification, the last vertex map $\lambda _{X}$ corresponds to a morphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) \rightarrow X$. Concretely, if $\tau $ is a $k$-simplex of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}})$ corresponding to a diagram then $\lambda _{X}(\tau )$ is the $k$-simplex of $X$ given by the composition where $f$ carries each vertex $\{ i\} \subseteq \Delta ^{k}$ to the image of the last vertex $\{ n_ i \} \subseteq \Delta ^{n_ i}$ under the map $\Delta ^{n_ i} \rightarrow \Delta ^{n_ k}$ given by horizontal composition in the diagram.
We can now state the main result of this section:
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.
Remark 3.3.4.9. Proposition 3.3.4.8 has a counterpart for the comparison map $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$, which we will prove in ยง3.3.5 (see Theorem 3.3.5.1).
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 3.2.8.3), 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
where the lower horizontal map is a homotopy equivalence by virtue of our inductive hypothesis (together with Proposition 3.1.6.17). It will therefore suffice to show that, for every morphism of simplicial sets $f: Y \rightarrow Q$, the induced map of fibers
is a homotopy equivalence (Proposition 3.2.8.1).
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 1.1.4.12 then supplies a pushout diagram of simplicial sets
which we can use to identify $\theta _{f}$ with the induced map
Invoking Proposition 3.2.8.1 again, we are reduced to showing that the horizontal maps appearing in the diagram
are homotopy equivalences. By virtue of Proposition 3.1.6.17, 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 3.1.6.20 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 3.2.4.2 (they can be realized as the nerves of partially ordered sets $\operatorname{Chain}[n]$ and $[n]$, each of which has a largest element). $\square$