Notation 3.3.2.1. Let $Q$ be a partially ordered set. We let $\operatorname{Chain}[Q]$ denote the collection of all nonempty, finite, linearly ordered subsets of $Q$. We regard $\operatorname{Chain}[Q]$ as a partially ordered set, where the partial order is given by inclusion. In the special case where $Q = [n] = \{ 0 < 1 < \ldots < n \} $ for some nonnegative integer $n$, we denote the partially ordered set $\operatorname{Chain}[Q]$ by $\operatorname{Chain}[n]$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
3.3.2 The Subdivision of a Simplex
Let $n \geq 0$ be a nonnegative integer and let
\[ | \Delta ^{n} | = \{ (t_0, t_1, \ldots , t_ n) \in [0,1]^{n+1} : t_0 + t_1 + \cdots + t_ n = 1 \} \]
be the topological $n$-simplex. For every nonempty subset $S \subseteq [n] = \{ 0 < 1 < \cdots < n \} $, let $| \Delta ^{S}|$ denote the corresponding face of $| \Delta ^{n} |$, given by the collection of tuples $(t_0, \ldots , t_ n) \in | \Delta ^{n} |$ satisfying $t_{i} =0$ for $i \notin S$. Let $b_{S}$ denote the barycenter of the simplex $| \Delta ^{S} |$: that is, the point $(t_0, \ldots , t_ n) \in | \Delta ^{S} | \subseteq | \Delta ^ n |$ given by $t_{i} = \begin{cases} \frac{1}{|S|} & \text{ if } i \in S \\ 0 & \text{ otherwise.} \end{cases}$ The collection of barycenters $\{ b_ S \} _{\emptyset \neq S \subseteq [n]}$ can be regarded as the vertices of a triangulation of $| \Delta ^{n} |$, which we indicate in the case $n = 2$ by the following diagram:
\[ \xymatrix@R =50pt@C=50pt{ & & \bullet \ar@ {-}[ddl] \ar@ {-}[ddr] \ar@ {-}[ddd] & & \\ & & & & \\ & \bullet \ar@ {-}[dr] & & \bullet \ar@ {-}[dl] & \\ & & \bullet & & \\ \bullet \ar@ {-}[urr] \ar@ {-}[rr] \ar@ {-}[uur] & & \bullet \ar@ {-}[u] & & \bullet \ar@ {-}[ll] \ar@ {-}[ull] \ar@ {-}[uul] } \]
In this section, we show that this triangulation arises from the identification of $| \Delta ^{n} |$ with the geometric realization of another simplicial set (Proposition 3.3.2.3).
Remark 3.3.2.2 (Functoriality). Let $f: Q \rightarrow Q'$ be a nondecreasing map between partially ordered sets. Then $f$ induces a map $\operatorname{Chain}[f]: \operatorname{Chain}[Q] \rightarrow \operatorname{Chain}[Q']$, which carries each nonempty linearly ordered subset $S \subseteq Q$ to its image $f(S) \subseteq Q'$. By means of this construction, we can regard $Q \mapsto \operatorname{Chain}[Q]$ as functor from the category of partially ordered sets to itself.
Proposition 3.3.2.3. Let $n \geq 0$ be an integer. Then there is a unique homeomorphism of topological spaces with the following properties:
For every nonempty subset $S \subseteq [n]$, the map $f$ carries $S$ (regarded as a vertex of $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$) to the barycenter $b_{S} \in | \Delta ^{S} | \subseteq | \Delta ^{n} |$.
For every $m$-simplex $\sigma : \Delta ^{m} \rightarrow \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$, the composite map
is affine: that is, it extends to an $\operatorname{\mathbf{R}}$-linear map from $\operatorname{\mathbf{R}}^{m+1} \supseteq | \Delta ^{m} |$ to $\operatorname{\mathbf{R}}^{n+1} \supseteq | \Delta ^{n} |$.
\[ f: | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow | \Delta ^{n} | \]
\[ | \Delta ^{m} | \xrightarrow { | \sigma | } | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \xrightarrow {f} | \Delta ^{n} | \]
Proof. Note that an affine map $| \Delta ^{m} | \rightarrow | \Delta ^{n} |$ is uniquely determined by its values on the vertices of the topological $m$-simplex $| \Delta ^{m} |$. From this observation, it is easy to deduce that there is a unique continuous function $f: | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow | \Delta ^{n} |$ which satisfies conditions $(1)$ and $(2)$ of Proposition 3.3.2.3. We will complete the proof by showing that $f$ is a homeomorphism. Since the domain and codomain of $f$ are compact Hausdorff spaces, it will suffice to show that $f$ is a bijection. Unwinding the definitions, this can be restated as follows:
- $(\ast )$
For every point $(t_0, t_1, \ldots , t_ n) \in | \Delta ^{n} |$, there exists a unique chain $S_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_ m$ of subsets of $[n]$ and positive real numbers $(s_0, s_1, \ldots , s_ m)$ satisfying the identities
\[ \sum s_ i = 1 \quad \quad (t_0, t_1, \ldots , t_ n) = \sum s_ i b_{S_{i}}. \]
We will deduce $(\ast )$ from the following more general assertion:
- $(\ast ')$
For every element $(t_0, t_1, \ldots , t_ n) \in \operatorname{\mathbf{R}}_{\geq 0}^{n+1}$, there exists a unique (possibly empty) chain $S_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_ m$ of subsets of $[n]$ and positive real numbers $(s_0, s_1, \ldots , s_ m)$ satisfying $(t_0, t_1, \ldots , t_ n) = \sum s_ i b_{S_{i}}.$
Note that, if $(t_0, t_1, \ldots , t_ n)$ and $(s_0, s_1, \ldots , s_ m)$ are as in $(\ast ')$, then we automatically have $\sum _{i=0}^{m} s_ i = \sum _{j=0}^{n} t_ j$. It follows that assertion $(\ast )$ is a special case of $(\ast ')$. To prove $(\ast ')$, let $K \subseteq [n]$ be the collection of those integers $j$ for which $t_ j \neq 0$. We proceed by induction on the cardinality of $k = |K|$. If $k=0$ is empty, there is nothing to prove. Otherwise, set $r = \min \{ k t_ i \} _{i \in K}$. We can then write
\[ (t_0, t_1, \ldots , t_ n) = r b_{K} + (t'_0, t'_1, \ldots , t'_ n) \]
for a unique sequence of nonnegative real numbers $(t'_0, \ldots , t'_ n )$. Applying our inductive hypothesis to the sequence $(t'_0, \ldots , t'_ n)$, we deduce that there is a unique chain of subsets $S_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_{m-1}$ of $[n]$ and positive real numbers $(s_0, s_1, \ldots , s_{m-1} )$ satisfying $(t'_0, t'_1, \ldots , t'_ n) = \sum s_ i b_{S_{i}}$. Note that each $S_{i}$ is contained in $K'$, and therefore properly contained in $K$. To complete the proof, we extend this sequence by setting $S_ m = K$ and $s_ m = r$. $\square$
Remark 3.3.2.4 (Functoriality). Let $\alpha : [m] \rightarrow [n]$ be a nondecreasing function between partially ordered sets, so that $\alpha $ induces a nondecreasing map $\operatorname{Chain}[\alpha ]: \operatorname{Chain}[m] \rightarrow \operatorname{Chain}[n]$ (Remark 3.3.2.2). If $\alpha $ is injective, then the diagram of topological spaces is commutative, where the horizontal maps are the homeomorphisms supplied by Proposition 3.3.2.3. Beware that if $\alpha $ is not injective, this this diagram does not necessarily commute. For example, the induced map $| \Delta ^{m} | \rightarrow | \Delta ^{n} |$ carries the barycenter of $| \Delta ^{m} |$ to the point which need not be the barycenter of any face $| \Delta ^{n} |$.
\[ \xymatrix@R =50pt@C=50pt{ | \operatorname{N}_{\bullet }( \operatorname{Chain}[m] ) | \ar [d]^{ | \operatorname{N}_{\bullet }( \operatorname{Chain}[\alpha ] ) |} \ar [r]^-{ f_ m }_-{\sim } & | \Delta ^{m} | \ar [d]^{ | \alpha | } \\ | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \ar [r]^-{ f_ n }_-{\sim } & | \Delta ^{n} | } \]
\[ ( \frac{ | \alpha ^{-1}(0) |}{m+1}, \frac{| \alpha ^{-1}(1) |}{m+1}, \ldots , \frac{| \alpha ^{-1}(n) |}{m+1}) \in | \Delta ^{n} |, \]
It will be convenient to repackage Proposition 3.3.2.3 (and Remark 3.3.2.4) as a statement about the singular simplicial set functor $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 1.2.2.2. We first introduce a bit of notation (which will play an essential role throughout ยง3.3).
Construction 3.3.2.5 (The $\operatorname{Ex}$ Functor). Let $X$ be a simplicial set. For every nonnegative integer $n$, we let $\operatorname{Ex}_{n}( X)$ denote the collection of all morphisms of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \rightarrow X$. By virtue of Remark 3.3.2.2, the construction $( [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto (\operatorname{Ex}_{n}(X) \in \operatorname{Set})$ determines a simplicial set which we will denote by $\operatorname{Ex}(X)$. The construction $X \mapsto \operatorname{Ex}(X)$ determines a functor from the category of simplicial sets to itself, which we denote by $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$.
Remark 3.3.2.6. The construction $X \mapsto \operatorname{Ex}(X)$ can be regarded as a special case of Variant 1.2.2.8: it is the functor $\operatorname{Sing}_{\bullet }^{T}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ associated to the cosimplicial object $T$ of $\operatorname{Set_{\Delta }}$ given by the construction $[n] \mapsto \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$.
Remark 3.3.2.7. The functor $X \mapsto \operatorname{Ex}(X)$ preserves filtered colimits of simplicial sets. To prove this, it suffices to observe that each of the simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ has only finitely many nondegenerate simplices (since the partially ordered set $\operatorname{Chain}[n])$ is finite).
Example 3.3.2.8. Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the nerve of $\operatorname{\mathcal{C}}$. Then $n$-simplices of the simplicial set $\operatorname{Ex}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$ can be identified with functors from the partially ordered set $\operatorname{Chain}[n]$ into $\operatorname{\mathcal{C}}$ (see Proposition 1.3.3.1).
Example 3.3.2.9. Let $X$ be a topological space and let $\operatorname{Sing}_{\bullet }(X)$ denote the singular simplicial set of $X$. For each nonnegative integer $n$, the $n$-simplices of $\operatorname{Sing}_{\bullet }(X)$ are given by continuous functions $| \Delta ^{n} | \rightarrow X$, and the $n$-simplices of $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$ are given by continuous functions $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow X$. The homeomorphism $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \simeq | \Delta ^{n} |$ of Proposition 3.3.2.3 determines a bijection $\operatorname{Sing}_{n}(X) \xrightarrow {\sim } \operatorname{Ex}_{n}( \operatorname{Sing}_{\bullet }(X) )$, and Remark 3.3.2.4 guarantees that these bijections are compatible with the face operators on the simplicial sets $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$. In other words, Proposition 3.3.2.3 supplies an isomorphism of semisimplicial sets $\varphi : \operatorname{Sing}_{\bullet }(X) \xrightarrow {\sim } \operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$. Beware that $\varphi $ is generally not an isomorphism of simplicial sets: that is, it usually does not commute with the degeneracy operators on $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$.
Variant 3.3.2.10 ($\operatorname{Ex}$ for Semisimplicial Sets). Note that, for every nonnegative integer $n$, the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ is braced (Exercise 3.3.1.2). If $X$ is a semisimplicial set, we write $\operatorname{Ex}_{n}(X)$ for the collection of all morphisms of semisimplicial sets $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )^{\mathrm{nd}} \rightarrow X$; here $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )^{\mathrm{nd}}$ denotes the semisimplicial subset of $\operatorname{N}_{\bullet }(\operatorname{Chain}[n] )$ spanned by the nondegenerate simplices. The construction $[n] \mapsto \operatorname{Ex}_{n}(X)$ determines a semisimplicial set, which we denote by $\operatorname{Ex}(X)$. Note that, if $X$ is the underlying semisimplicial set of a simplicial set $Y$, then $\operatorname{Ex}(X)$ is the underlying semisimplicial set of the simplicial set $\operatorname{Ex}(Y)$ given by Construction 3.3.2.5 (this is a special case of Proposition 3.3.1.5). In other words, the construction $X \mapsto \operatorname{Ex}(X)$ determines a functor from the category of semisimplicial sets to itself which fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Simplicial sets} \} \ar [r]^-{\operatorname{Ex}} \ar [d] & \{ \text{Simplicial sets} \} \ar [d] \\ \{ \text{Semisimplicial sets} \} \ar [r]^-{\operatorname{Ex}} & \{ \text{Semisimplicial sets} \} . } \]