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6.3.7 Subdivision and Localization

Our goal in this section is to prove the following:

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 $\operatorname{N}_{\bullet }(A) \rightarrow S$.

Our proof of Theorem 6.3.7.1 will make use of the subdivision construction introduced in ยง3.3.3.

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 6.3.6.4, 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 6.3.7.2.

Proof of Proposition 6.3.7.2. By virtue of Proposition 6.3.6.10, 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 1.1.3.13, 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 6.3.6.11 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 $P \subseteq [n]$. We wish to show that, for every morphism of simplicial sets $\alpha : T \rightarrow S$, the projection map $\pi : T \times _{S} \operatorname{Sd}(S) \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} \operatorname{Sd}(S)$ with respect to some collection of edges. By virtue of Proposition 6.3.6.2, we can assume without loss of generality that $T = \Delta ^{m}$ is a standard simplex, so that $\alpha $ can be identified with a nondecreasing map of linearly ordered sets $[m] \rightarrow [n]$. Unwinding the definitions, we can identify $T \times _{S} \operatorname{Sd}(S)$ with the nerve of the partially ordered set $A \subseteq [m] \times \operatorname{Chain}[n]$ consisting of those pairs $(i, P)$ satisfying $\max (P) = \alpha (i)$. Under this identification, the projection map $\pi $ is induced by the morphism of partially ordered sets

\[ A \rightarrow [m] \quad \quad (i,P) \mapsto i. \]

It follows that $\pi $ is a reflective localization: it has a fully faithful right adjoint, given by the construction $i \mapsto (i, \{ 0 < 1 < \cdots < \alpha (i) \} )$. The desired result is now a consequence of Proposition 6.3.3.9. $\square$

Using Proposition 6.3.7.2, we can immediately deduce that Theorem 6.3.7.1 holds for a large class of simplicial sets $S$.

Definition 6.3.7.4. Let $S$ be a simplicial set. We say that $S$ is nonsingular if, for every every nondegenerate $n$-simplex $\sigma $ of $S$, the corresponding map $\sigma : \Delta ^ n \rightarrow S$ is a monomorphism of simplicial sets.

Remark 6.3.7.5. 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 6.3.7.6. 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 6.3.7.7. Let $S$ be a nonsingular simplicial set. Then every simplicial subset $S' \subseteq S$ is also nonsingular.

Remark 6.3.7.8. 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 6.3.7.9. 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 6.3.7.10. Let $S$ be a simplicial set, and let $\operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S)$ denotes the collection of simplicial subsets $K \subseteq S$ which are isomorphic to a standard simplex. We regard $\operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S)$ as a partially ordered set with respect to inclusion. If $S$ is nonsingular, the construction

\[ ( \sigma : \Delta ^ n \rightarrow S) \mapsto ( \operatorname{im}(\sigma ) \subseteq S ) \]

determines an isomorphism of categories $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \simeq \operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S)$, where $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}}$ denotes the category of nondegenerate simplices of $S$ (Notation 3.3.3.9). Combining this observation with Proposition 3.3.3.15, we obtain an isomorphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S) ) \simeq \operatorname{Sd}(S)$.

Corollary 6.3.7.11. Let $S$ be a nonsingular simplicial set. Then the last vertex map determines a universally localizing morphism $\operatorname{N}_{\bullet }( \operatorname{Sub}_{\operatorname{{\bf \Delta }}}(S) ) \rightarrow S$.

For our purposes, Corollary 6.3.7.11 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 6.3.7.11 with the following result:

Proposition 6.3.7.12. 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 6.3.7.12 will make use of the following:

Lemma 6.3.7.13. Let $\{ S_{\alpha } \} $ be a diagram of simplicial sets. Then the limit $\varprojlim _{\alpha } S_{\alpha }$ is also nonsingular.

Proof. By virtue of Remark 6.3.7.7, 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 1.1.3.4 guarantees that there exists a commutative diagram

\[ \xymatrix { \Delta ^{n} \ar [r]^{\sigma } \ar [d]^{\tau _{\alpha }} & S \ar [d] \\ \Delta ^{n_{\alpha }} \ar [r]^{ \sigma _{\alpha } } & S_{\alpha }, } \]

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 6.3.7.6). It will therefore suffice to show that $\tau $ is nondegenerate, which follows immediately from our assumption that $\sigma $ is nondegenerate. $\square$

Proof of Proposition 6.3.7.12. Let $S$ be a simplicial set. For each integer $k \geq 0$, let $\operatorname{sk}_{k}(S)$ denote the $k$-skeleton of $S$ (Construction 1.1.3.5). We will construct a commutative diagram

\[ \xymatrix { \widetilde{\operatorname{sk}}_{0}(S) \ar@ {^{(}->}[r] \ar [d]^{\varphi _0} & \widetilde{\operatorname{sk}}_{1}(S) \ar [d]^{ \varphi _1} \ar@ {^{(}->}[r] & \widetilde{\operatorname{sk}}_{2}(S) \ar [d]^{ \varphi _2} \ar@ {^{(}->}[r] & \cdots \\ \operatorname{sk}_0(S) \ar@ {^{(}->}[r] & \operatorname{sk}_1(S) \ar@ {^{(}->}[r] & \operatorname{sk}_2(S) \ar@ {^{(}->}[r] & \cdots } \]

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 6.3.7.8 that the colimit $\widetilde{S} = \varinjlim _{k} \widetilde{\operatorname{sk}}_{k}(S)$ is nonsingular. Applying Proposition 6.3.6.10, 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 1.1.3.13 supplies a pushout diagram

\[ \xymatrix { T_0 \ar [r] \ar [d] & T \ar [d] \\ \operatorname{sk}_{k-1}(S) \ar [r] & \operatorname{sk}_{k}(S). } \]

Note that $T$ is nonsingular (Example 6.3.7.6), so the simplicial subset $T_0 \subseteq T$ is also nonsingular (Remark 6.3.7.7). 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

6.6
\begin{equation} \begin{gathered}\label{equation:resolution-of-sset} \xymatrix { \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 6.3.6.9 that the vertical maps in the diagram (6.6) are universally localizing. Applying Proposition 6.3.6.11, 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 6.3.7.8, 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 6.3.7.6), we are reduced to proving that the cone $\widetilde{T}_{0}^{\triangleright }$ is nonsingular (Lemma 6.3.7.13). By virtue of Remark 6.3.7.9, we can reduce further to showing that $\widetilde{T}_{0}$ is nonsingular. This follows from Remark 6.3.7.7 and Lemma 6.3.7.13, since $\widetilde{T}_0$ can be identified with a simplicial subset of the product $T \times \widetilde{\operatorname{sk}}_{k-1}(S)$. $\square$

Remark 6.3.7.14. 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 6.3.7.12 will also be finite. Specializing to the case $k \geq \dim (S)$, we obtain a universally localizing morphism

\[ \widetilde{\operatorname{sk}}_{k}(S) \rightarrow \operatorname{sk}_{k}(S) = S \]

where the simplicial set $\widetilde{\operatorname{sk}}_{k}(S)$ is both finite and nonsingular.

Proof of Theorem 6.3.7.1. Let $S$ be a simplicial set. Applying Proposition 6.3.7.12, 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 6.3.7.11 supplies a universally localizing morphism $\lambda _{ \widetilde{S} }: \operatorname{N}_{\bullet }(A) \rightarrow \widetilde{S}$. Applying Proposition 6.3.6.8, 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 6.3.7.14, we also obtain the following:

Variant 6.3.7.15. 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 6.3.7.16. 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 $\widetilde{S} \twoheadrightarrow S$.