Theorem 6.3.7.1. Let $S$ be a simplicial set. Then there exists a partially ordered set $(A,\leq )$ and a universally localizing morphism $\operatorname{N}_{\bullet }(A) \rightarrow S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
6.3.7 Subdivision and Localization
Our goal in this section is to prove the following:
We begin by proving a weaker version of Theorem 6.3.7.1, which asserts that every simplicial set $S$ admits a universally localizing morphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow S$, for some category $\operatorname{\mathcal{C}}$. Here it is possible to be completely explicit: we can take $\operatorname{\mathcal{C}}$ to be the category of simplices $\operatorname{{\bf \Delta }}_{S}$ introduced in Construction 1.1.3.9 (Corollary 6.3.7.5).
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.
Remark 6.3.7.3. Let $S$ be a simplicial set. Combining Proposition 6.3.7.2 with Remark 6.3.6.5, we recover the assertion that the last vertex map $\lambda _{S}: \operatorname{Sd}(S) \rightarrow S$ is a weak homotopy equivalence. In other words, we can regard Proposition 6.3.7.2 as a refinement of Proposition 3.3.4.8.
Proof of Proposition 6.3.7.2. By virtue of Proposition 6.3.6.12, 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 1.1.4.12, 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 6.3.6.13 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 $P \subseteq [n]$. Under this identification, $\lambda _{S}$ is obtained from the map of partially ordered sets
\[ \operatorname{Chain}[n] \rightarrow [n] \quad \quad (S \subseteq [n] ) \mapsto \mathrm{max}(S). \]
We now observe that this map admits section $\mu : [n] \rightarrow \operatorname{Chain}[n]$, given by the construction $i \mapsto \{ 0 < 1 < \cdots < i \} $, and there is a (unique) natural transformation $\operatorname{id}_{ \operatorname{Sd}(S) } \rightarrow \mu \circ \lambda _{S}$ which belongs to $\operatorname{Fun}_{/S}( \operatorname{Sd}(S), \operatorname{Sd}(S) )$. The desired result now follows from the criterion of Proposition 6.3.6.8. $\square$
Variant 6.3.7.4. Let $S$ be a simplicial set and let $\psi _{S}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow \operatorname{Sd}(S)$ be the comparison map of Construction 3.3.3.10. Then $\psi _{S}$ is universally localizing.
Proof. Note that the functor $S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ preserves small colimits (Variant 3.3.3.20). Proceeding as in the proof of Proposition 6.3.7.2, we can reduce to the case where $S = \Delta ^{n}$ is a standard simplex. In this case, we can identify $\psi _{S}$ with (the nerve of) the functor
\[ \operatorname{{\bf \Delta }}_{S} \rightarrow \operatorname{Chain}[n] \quad \quad (\alpha : [m] \rightarrow [n] \mapsto \mathrm{im}(\alpha ) \subseteq [n] ) \]
This functor admits a section $\phi $, which identifies $\operatorname{Chain}[n]$ with the subcategory $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \subseteq \operatorname{{\bf \Delta }}_{S}$ of nondegenerate simplices of $S$. Note that there is a (unique) natural transformation from the identity functor $\operatorname{id}_{ \operatorname{{\bf \Delta }}_{S} }$ to $\phi \circ \psi _{S}$ which belongs to $\operatorname{Fun}_{ / \operatorname{Sd}(S) }( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ), \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) )$, so the desired result follows from the criterion of Proposition 6.3.6.8. $\square$
Corollary 6.3.7.5. Let $S$ be a simplicial set. Then the composite morphism is universally localizing.
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \xrightarrow { \psi _{S} } \operatorname{Sd}(S) \xrightarrow { \lambda _{S} } S \]
Proof. By virtue of Proposition 6.3.6.10, this follows from the observation that the morphisms $\lambda _{S}$ and $\psi _{S}$ are universally localizing (Proposition 6.3.7.2 and Variant 6.3.7.4). $\square$
We now study some additional assumptions on the simplicial set $S$ which will allow us to replace the category $\operatorname{{\bf \Delta }}_{S}$ of Corollary 6.3.7.5 by a partially ordered set.
Definition 6.3.7.6. Let $S$ be a simplicial set. We say that $S$ is nonsingular if, for every nondegenerate $n$-simplex $\sigma $ of $S$, the corresponding map $\sigma : \Delta ^ n \rightarrow S$ is a monomorphism of simplicial sets.
Remark 6.3.7.7. Recall that a simplicial set $S$ is braced if the collection of nondegenerate simplices of $S$ is closed under the face operators (Definition 3.3.1.1). Every nonsingular simplicial set is braced. However, the converse is false. For example, the quotient $\Delta ^1 / \operatorname{\partial \Delta }^{1}$ is braced, but is not nonsingular.
Example 6.3.7.8. Let $(A, \leq )$ be a partially ordered set. Then the nerve $\operatorname{N}_{\bullet }(A)$ is a nonsingular simplicial set. In particular, for every integer $n \geq 0$, the standard simplex $\Delta ^{n}$ is nonsingular.
Remark 6.3.7.9. Let $S$ be a nonsingular simplicial set. Then every simplicial subset $S' \subseteq S$ is also nonsingular.
Remark 6.3.7.10. Let $S$ be a simplicial set which can be written as a union of a collection of simplicial subsets $\{ S_{\alpha } \subseteq S \} $. If each $S_{\alpha }$ is nonsingular, then $S$ is nonsingular.
Remark 6.3.7.11. Let $S$ and $T$ be nonsingular simplicial sets. Then the join $S \star T$ is nonsingular. In particular, if $S$ is nonsingular, then the cone $S^{\triangleright }$ is also nonsingular.
Remark 6.3.7.12. Let $S$ be a simplicial set and $A$ denote the collection of simplicial subsets $S' \subseteq S$ which are isomorphic to a standard simplex. We regard $A$ as a partially ordered set with respect to inclusion. If $S$ is nonsingular, the construction determines an isomorphism of categories $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \simeq A$, where $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}}$ denotes category of nondegenerate simplices of $S$ (Notation 3.3.3.12). Combining this observation with Proposition 3.3.3.17, we obtain an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{Sd}(S)$.
\[ ( \sigma : \Delta ^ n \rightarrow S) \mapsto ( \operatorname{im}(\sigma ) \subseteq S ) \]
Corollary 6.3.7.13. Let $S$ be a nonsingular simplicial set. Then there exists a partially ordered set $A$ and a universally localizing morphism $\operatorname{N}_{\bullet }(A) \rightarrow S$.
Proof. Combine Proposition 6.3.7.2 with Remark 6.3.7.12. $\square$
For our purposes, Corollary 6.3.7.13 is a poor replacement for Theorem 6.3.7.1: an $\infty $-category $\operatorname{\mathcal{C}}$ is rarely nonsingular when regarded as a simplicial set (see Exercise 3.3.1.2). We will deduce the general form of Theorem 6.3.7.1 by combining Corollary 6.3.7.13 with the following result:
Proposition 6.3.7.14. Let $S$ be a simplicial set. Then there exists a universally localizing morphism $\varphi : \widetilde{S} \rightarrow S$, where $\widetilde{S}$ is nonsingular.
The proof of Proposition 6.3.7.14 will make use of the following:
Lemma 6.3.7.15. Let $\{ S_{\alpha } \} $ be a diagram of nonsingular simplicial sets. Then the limit $\varprojlim _{\alpha } S_{\alpha }$ is also nonsingular.
Proof. By virtue of Remark 6.3.7.9, it will suffice to show that the product $S = \prod _{\alpha } S_{\alpha }$ is nonsingular. Let $\sigma : \Delta ^ n \rightarrow S$ be a nondegenerate simplex of $S$; we wish to show that $\sigma $ is a monomorphism of simplicial sets. For each index $\alpha $, Proposition 1.1.3.8 guarantees that there exists a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \ar [r]^-{\sigma } \ar [d]^{\tau _{\alpha }} & S \ar [d] \\ \Delta ^{n_{\alpha }} \ar [r]^-{ \sigma _{\alpha } } & S_{\alpha }, } \]
where $\sigma _{\alpha }$ is a nondegenerate simplex $S_{\alpha }$. Our assumption that $S_{\alpha }$ is nondegenerate guarantees that $\sigma _{\alpha }$ is a monomorphism of simplicial sets, so that the product map
\[ \prod _{\alpha } \Delta ^{n_{\alpha }} \xrightarrow { \prod _{\alpha } \sigma _{\alpha } } \prod _{\alpha } S_{\alpha } = S \]
is also a monomorphism. It will therefore suffice to show that $\tau = \{ \tau _{\alpha } \} $ determines a monomorphism of simplicial sets $\Delta ^{n} \rightarrow \prod _{\alpha } \Delta ^{n_{\alpha }}$. Since $\prod _{\alpha } \Delta ^{n_{\alpha }}$ can be identified with the nerve of the partially ordered set $\prod _{\alpha } [n_{\alpha } ]$, it is a nonsingular simplicial set (Example 6.3.7.8). It will therefore suffice to show that $\tau $ is nondegenerate, which follows immediately from our assumption that $\sigma $ is nondegenerate. $\square$
Proof of Proposition 6.3.7.14. Let $S$ be a simplicial set. For each integer $k \geq 0$, let $\operatorname{sk}_{k}(S)$ denote the $k$-skeleton of $S$ (Construction 1.1.4.1). We will construct a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{sk}}_{0}(S) \ar@ {^{(}->}[r] \ar [d]^{\varphi _0} & \widetilde{\operatorname{sk}}_{1}(S) \ar [d]^{ \varphi _1} \ar@ {^{(}->}[r] & \widetilde{\operatorname{sk}}_{2}(S) \ar [d]^{ \varphi _2} \ar@ {^{(}->}[r] & \cdots \\ \operatorname{sk}_0(S) \ar@ {^{(}->}[r] & \operatorname{sk}_1(S) \ar@ {^{(}->}[r] & \operatorname{sk}_2(S) \ar@ {^{(}->}[r] & \cdots } \]
where each of the horizontal maps is a monomorphism, each of the vertical maps is universally localizing, and each of the simplicial sets $\widetilde{\operatorname{sk}}_{k}(S)$ is nonsingular. It then follows from Remark 6.3.7.10 that the colimit $\widetilde{S} = \varinjlim _{k} \widetilde{\operatorname{sk}}_{k}(S)$ is nonsingular. Applying Proposition 6.3.6.12, we conclude that the morphisms $\varphi _{k}$ determine a universally localizing morphism $\varphi : \widetilde{S} \rightarrow S$.
The construction of the morphisms $\varphi _{k}: \widetilde{\operatorname{sk}}_{k}(S) \rightarrow \operatorname{sk}_{k}(S)$ proceeds by induction. If $k=0$, we can take $\widetilde{\operatorname{sk}}_{k}(S) = \operatorname{sk}_ k(S)$ and $\varphi _{k}$ to be the identity morphism. Let us therefore assume that $k > 0$, and that the morphism $\varphi _{k-1}: \widetilde{\operatorname{sk}}_{k-1}(S) \rightarrow \operatorname{sk}_{k-1}(S)$ has already been constructed. Let $S_{k}^{\mathrm{nd}}$ denote the set of nondegenerate $k$-simplices of $S$, let $T$ denote the coproduct $\coprod _{\sigma \in S_ k^{\mathrm{nd}}} \Delta ^{k}$, and let $T_0 \subseteq T$ denote the coproduct $\coprod _{\sigma \in S_{k}^{\mathrm{nd}}} \operatorname{\partial \Delta }^{k}$, so that Proposition 1.1.4.12 supplies a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ T_0 \ar [r] \ar [d] & T \ar [d] \\ \operatorname{sk}_{k-1}(S) \ar [r] & \operatorname{sk}_{k}(S). } \]
Note that $T$ is nonsingular (Example 6.3.7.8), so the simplicial subset $T_0 \subseteq T$ is also nonsingular (Remark 6.3.7.9). Let $\widetilde{T}_0$ denote the fiber product $T_0 \times _{ \operatorname{sk}_{k-1}(S)} \widetilde{\operatorname{sk}}_{k-1}(S)$, and we define $\widetilde{\operatorname{sk}}_{k}(S)$ to be the pushout of the diagram
\[ ( \widetilde{\operatorname{sk}}_{k-1}(S) \times \widetilde{T}_{0}^{\triangleright } ) \hookleftarrow \widetilde{T}_0 \hookrightarrow ( T \times \widetilde{T}_0^{\triangleright } ). \]
Note that the cone point of $\widetilde{T}_{0}^{\triangleright }$ determines an embedding $\widetilde{\operatorname{sk}}_{k-1}(S) \rightarrow \widetilde{\operatorname{sk}}_{k}(S)$. Moreover, we have a commutative diagram
6.16
\begin{equation} \begin{gathered}\label{equation:resolution-of-sset} \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{sk}}_{k-1}(S) \times \widetilde{T}_0^{\triangleright } \ar [d] & \widetilde{T}_0 \ar [l] \ar [r] \ar [d] & T \times \widetilde{T}_0^{\triangleright } \ar [d] \\ \operatorname{sk}_{k-1}(S) & T_0 \ar [l] \ar [r] & T. } \end{gathered} \end{equation}
which determines an extension of $\varphi _{k-1}$ to a map
\[ \varphi _{k}: \widetilde{\operatorname{sk}}_{k}(S) \rightarrow \operatorname{sk}_{k-1}(S) \coprod _{T_0} T \simeq \operatorname{sk}_{k}(S). \]
Since the cone $\widetilde{T}_0^{\triangleright }$ is weakly contractible, it follows from Corollary 6.3.6.11 that the vertical maps in the diagram (6.16) are universally localizing. Applying Proposition 6.3.6.13, we deduce that $\varphi _{k}$ is also universally localizing.
To complete the proof, it will suffice to show that the simplicial set $\widetilde{\operatorname{sk}}_{k}(S)$ is nonsingular. By virtue of Remark 6.3.7.10, it will suffice to show that the simplicial subsets $\widetilde{\operatorname{sk}}_{k-1}(S) \times \widetilde{T}_0^{\triangleright }$ and $T \times \widetilde{T}_0^{\triangleright }$ are nonsingular. Since $\widetilde{\operatorname{sk}}_{k-1}(S)$ is nonsingular (by our inductive hypothesis) and $T$ is nonsingular (Example 6.3.7.8), we are reduced to proving that the cone $\widetilde{T}_{0}^{\triangleright }$ is nonsingular (Lemma 6.3.7.15). By virtue of Remark 6.3.7.11, we can reduce further to showing that $\widetilde{T}_{0}$ is nonsingular. This follows from Remark 6.3.7.9 and Lemma 6.3.7.15, since $\widetilde{T}_0$ can be identified with a simplicial subset of the product $T \times \widetilde{\operatorname{sk}}_{k-1}(S)$. $\square$
Remark 6.3.7.16. Let $S$ be a finite simplicial set. In this case, each of the simplicial sets $\widetilde{\operatorname{sk}}_{k}(S)$ constructed in the proof of Proposition 6.3.7.14 will also be finite. Specializing to the case $k \geq \dim (S)$, we obtain a universally localizing morphism where the simplicial set $\widetilde{\operatorname{sk}}_{k}(S)$ is both finite and nonsingular.
\[ \widetilde{\operatorname{sk}}_{k}(S) \rightarrow \operatorname{sk}_{k}(S) = S \]
Proof of Theorem 6.3.7.1. Let $S$ be a simplicial set. Applying Proposition 6.3.7.14, we can choose a universally localizing morphism $\varphi : \widetilde{S} \rightarrow S$, where $\widetilde{S}$ is a nonsingular simplicial set. Let $A = \operatorname{Sub}_{\operatorname{{\bf \Delta }}}( \widetilde{S} )$ denote the partially ordered set of simplicial subsets of $\widetilde{S}$ which are isomorphic to a standard simplex, so that Corollary 6.3.7.13 supplies a universally localizing morphism $\lambda _{ \widetilde{S} }: \operatorname{N}_{\bullet }(A) \rightarrow \widetilde{S}$. Applying Proposition 6.3.6.10, we deduce that the composite morphism
\[ \operatorname{N}_{\bullet }(A) \xrightarrow { \lambda _{ \widetilde{S} } } \widetilde{S} \xrightarrow { \varphi } S \]
is also universally localizing. $\square$
Combining the preceding argument with Remark 6.3.7.16, we also obtain the following:
Variant 6.3.7.17. Let $S$ be a finite simplicial set. Then there exists a finite partially ordered set $(A,\leq )$ and a universally localizing morphism $\operatorname{N}_{\bullet }(A) \rightarrow S$.
Exercise 6.3.7.18. Let $S$ be a simplicial set and let $\widetilde{S}$ be the smallest simplicial subset of $S \times \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ which contains all simplices of the form $(\sigma , \tau )$, where $\tau $ is a nondegenerate simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ (that is, it corresponds to a strictly increasing sequence of nonnegative integers). Show that $\widetilde{S}$ is nonsingular, and that projection onto the first factor determines a universally localizing morphism $\widetilde{S} \twoheadrightarrow S$.