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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).

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 $\mathrm{Max}: Q \rightarrow \operatorname{Chain}(Q)$, where $\operatorname{Chain}(Q)$ is defined as in Notation 3.3.1.1.

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Chain}(P) \ar [r]^-{\mathrm{Max}} \ar [d]^{ S \mapsto f(S) } & P \ar [d]^{f} \\ \operatorname{Chain}(Q) \ar [r]^-{ \mathrm{Max} } & Q } \]

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

\[ \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \xrightarrow { \mathrm{Max} } \Delta ^ n \xrightarrow {\sigma } X, \]

which we regard as an $n$-simplex of the simplicial set $\operatorname{Ex}(X)$ of Construction 3.3.1.5. 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

\[ \xymatrix { X \ar [r]^-{ \rho _ X } \ar [d]^{f} & \operatorname{Ex}(X) \ar [d]^{ \operatorname{Ex}(f) } & \operatorname{Sd}(X) \ar [r]^-{ \lambda _ X} \ar [d]^{ \operatorname{Sd}(f) } & X \ar [d]^{f} \\ Y \ar [r]^-{ \rho _ Y } & \operatorname{Ex}(Y) & \operatorname{Sd}(Y) \ar [r]^-{ \lambda _ Y } & Y } \]

are commutative. We may therefore regard the constructions $X \mapsto \rho _{X}$ and $X \mapsto \lambda _{X}$ as natural transformations of functors

\[ \rho : \operatorname{id}_{ \operatorname{Set_{\Delta }}} \rightarrow \operatorname{Ex}\quad \quad \lambda : \operatorname{Sd}\rightarrow \operatorname{id}_{ \operatorname{Set_{\Delta }}}. \]

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 3.3.3.16). 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 1.1.4.9). Then the maps

\[ \rho _{X}: X \rightarrow \operatorname{Ex}(X) \quad \quad \lambda _{X}: \operatorname{Sd}(X) \rightarrow X \]

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 of $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ of nondegenerate simplices of $X$ (Proposition 3.3.3.15). 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

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n_0} \ar [r] \ar [drr]^{ \sigma _0 } & \Delta ^{n_1} \ar [dr]^{ \sigma _1} & \cdots \ar [r] & \Delta ^{n_{k-1}} \ar [dl]^{ \sigma _{k-1} } \ar [r] & \Delta ^{n_ k} \ar [dll]^{\sigma _{k}} \\ & & X, & & } \]

then $\lambda _{X}(\tau )$ is the $k$-simplex of $X$ given by the composition

\[ \Delta ^{k} \xrightarrow { f } \Delta ^{n_ k} \xrightarrow { \sigma _ k} X, \]

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 3.2.7.3), 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 3.1.7.5). 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 1.1.3.11 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 3.1.7.5, 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 3.1.5.14 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). $\square$