Definition 3.3.3.1 (Subdivision). Let $X$ and $Y$ be simplicial sets. We will say that a morphism of simplicial sets $u: X \rightarrow \operatorname{Ex}(Y)$ exhibits $Y$ as a subdivision of $X$ if, for every simplicial set $Z$, composition with $u$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, Z ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{Ex}(Z) )$ (see Construction 3.3.2.5).
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3.3.3 The Subdivision of a Simplicial Set
Let $n \geq 0$ be a nonnegative integer. In §3.3.2, we showed that the topological $n$-simplex $| \Delta ^{n} |$ can be identified with the geometric realization of the set of its faces $\operatorname{Chain}[n]$, partially ordered by inclusion (Proposition 3.3.2.3). We now prove a generalization of this result, replacing the standard simplex $\Delta ^{n}$ by an arbitrary braced simplicial set $X$ and the nerve $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ by another simplicial set $\operatorname{Sd}(X)$, which we will refer to as the subdivision of $X$.
Notation 3.3.3.2. Let $X$ be a simplicial set. It follows immediately from the definitions that if there exists a simplicial set $Y$ and a morphism $u: X \rightarrow \operatorname{Ex}(Y)$ which exhibits $Y$ as a subdivision of $X$, then the simplicial set $Y$ (and the morphism $u$) are uniquely determined up to isomorphism and depend functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{Sd}(X)$ and refer to it as the subdivision of $X$.
Proposition 3.3.3.3. Let $X$ be a simplicial set. Then there exists another simplicial set $\operatorname{Sd}(X)$ and a morphism $u: X \rightarrow \operatorname{Ex}( \operatorname{Sd}(X) )$ which exhibits $\operatorname{Sd}(X)$ as a subdivision of $X$, in the sense of Notation 3.3.3.2.
Proof. By virtue of Remark 3.3.2.6, this is a special case of Proposition 1.2.3.15. $\square$
Corollary 3.3.3.4. The functor $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint, given by the construction $X \mapsto \operatorname{Sd}(X)$.
Example 3.3.3.5. Let $n$ be a nonnegative integer. Then the identity map determines a map of simplicial sets $u: \Delta ^ n \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) )$, which exhibits $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ as the subdivision of $\Delta ^ n$. In particular, the subdivision $\operatorname{Sd}(\Delta ^2)$ is the $2$-dimensional simplicial set indicated in the diagram
\[ \operatorname{id}: \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \]
\[ \xymatrix@R =50pt@C=50pt{ & & \{ 1\} \ar [ddl] \ar [ddr] \ar [ddd] & & \\ & & & & \\ & \{ 0,1\} \ar [dr] & & \{ 1,2\} \ar [dl] & \\ & & \{ 0,1,2\} & & \\ \{ 0\} \ar [urr] \ar [rr] \ar [uur] & & \{ 0,2 \} \ar [u] & & \{ 2\} . \ar [ll] \ar [ull] \ar [uul] } \]
Proposition 3.3.3.6. Let $X$ be a braced simplicial set. Then there is a canonical homeomorphism of topological spaces $f_ X: | \operatorname{Sd}(X) | \rightarrow |X|$.
Proof. For every topological space $Y$, Example 3.3.2.9 supplies an isomorphism of semisimplicial sets $\operatorname{Sing}_{\bullet }(Y) \rightarrow \operatorname{Ex}( \operatorname{Sing}_{\bullet }(Y) )$. These isomorphisms depend functorially on $Y$, and can therefore be regarded as an isomorphism of functors $G \circ \operatorname{Sing}_{\bullet } \xrightarrow {\sim } G \circ \operatorname{Ex}\circ \operatorname{Sing}_{\bullet }$, where $G: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}, \operatorname{Set})$ denotes the forgetful functor from simplicial sets to semisimplicial sets. Passing to left adjoints, we conclude that for every semisimplicial set $S_{\bullet }$, we have a canonical homeomorphism $| \operatorname{Sd}( S_{\bullet }^{+} ) | \simeq | S_{\bullet }^{+} |$, depending functorially on $S_{\bullet }$. Proposition 3.3.3.6 now follows from Corollary 3.3.1.11 (applied to the semisimplicial set $X^{\mathrm{nd}}$). $\square$
Remark 3.3.3.7. The homeomorphisms $f_{X}: | \operatorname{Sd}(X) | \simeq |X|$ constructed in the proof of Proposition 3.3.3.6 are characterized by the following properties:
In the special case where $X = \Delta ^ n$ is a standard simplex, $f_{X}$ is given by the composition
where the first map is supplied by the identification $\operatorname{Sd}( \Delta ^{n} ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ of Example 3.3.3.5 and $f$ is the homeomorphism of Proposition 3.3.2.3.
Let $u: X \rightarrow Y$ be a morphism of braced simplicial sets which carries nondegenerate simplices of $X$ to nondegenerate simplices of $Y$. Then the diagram of topological spaces
commutes.
\[ | \operatorname{Sd}( \Delta ^ n) | \simeq | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \xrightarrow {f} | \Delta ^ n |, \]
\[ \xymatrix@R =50pt@C=50pt{ | \operatorname{Sd}(X) | \ar [r]^-{ f_ X }_-{\sim } \ar [d]^{ | \operatorname{Sd}(u) |} & | X | \ar [d]^{ | u | } \\ | \operatorname{Sd}(Y) | \ar [r]^-{ f_ Y}_-{\sim } & | Y | } \]
Warning 3.3.3.8. Let $u: X \rightarrow Y$ be a morphism of braced simplicial sets. If $u$ does not carry nondegenerate simplices of $X$ to nondegenerate simplices of $Y$, then the diagram of topological spaces does not necessarily commute (this phenomenon occurs already in the case where $X$ and $Y$ are simplices: see Remark 3.3.2.4).
\[ \xymatrix@R =50pt@C=50pt{ | \operatorname{Sd}(X) | \ar [r]^-{ f_ X } \ar [d]^{ | \operatorname{Sd}(u) |}_-{\sim } & | X | \ar [d]^{ | u | } \\ | \operatorname{Sd}(Y) | \ar [r]^-{ f_ Y}_-{\sim } & | Y | } \]
The subdivision construction is closely related to the category of simplices introduced in §1.2.3.
Construction 3.3.3.9. Let $X$ be a simplicial set and let $\operatorname{{\bf \Delta }}_{X}$ denote the category of simplices of $X$ (Construction 1.1.3.9). Unwinding the definitions, we see that $n$-simplices $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} )$ can be identified with diagrams of simplicial sets For $0 \leq i \leq n$, let $S_{i} \subseteq [k_ n ]$ be the image of the underlying map of linearly ordered sets $[k_0] \rightarrow [k_ n]$, and suppose we are given a morphism $u: X \rightarrow \operatorname{Ex}(Y)$ which exhibits $Y$ as a subdivision of $X$. Then $u$ carries $\tau $ to a $k_{n}$-simplex of $\operatorname{Ex}(Y)$, which we can identify with a morphism $\operatorname{N}_{\bullet }( \operatorname{Chain}[k_ n] ) \rightarrow Y$ carrying $( S_0 \subseteq S_1 \subseteq \cdots \subseteq S_ n )$ to an $n$-simplex $\sigma '$ of $Y$. The construction $\sigma \mapsto \sigma '$ depends functorially on $[n]$, and therefore determines a comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow Y = \operatorname{Sd}(X)$.
\[ \Delta ^{k_0} \rightarrow \Delta ^{k_1} \rightarrow \cdots \rightarrow \Delta ^{k_ n} \xrightarrow {\tau } X. \]
Example 3.3.3.10. Let $X = \Delta ^ n$ be a standard simplex. Then the comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.9, can be identified with the nerve of the functor $\operatorname{{\bf \Delta }}_{X} \rightarrow \operatorname{Chain}[n]$, which carries each morphism $\Delta ^{m} \rightarrow \Delta ^{n}$ to the image of the underlying map of linearly ordered sets $[m] \rightarrow [n]$.
Notation 3.3.3.11. Let $X$ be a simplicial set and let $\operatorname{{\bf \Delta }}_{X}$ be the category of simplices of $X$ (Construction 1.1.3.9). By definition, the objects of $\operatorname{{\bf \Delta }}_{X}$ are given by pairs $([n], \sigma )$, where $n$ is a nonnegative integer and $\sigma $ is an $n$-simplex of $X$. We let $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{X}$ spanned by those pairs $([n], \sigma )$ where $\sigma $ is a nondegenerate $n$-simplex of $X$. We will refer to $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ as the category of nondegenerate simplices of $X$.
Example 3.3.3.12. Let $S$ be a semisimplicial set, and let $S^{+}$ be the braced simplicial set given by Construction 3.3.1.6. Then the category of nondegenerate simplices $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ can be described concretely as follows:
The objects of $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ are pairs $( [n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ and $\sigma $ is an element of $S_{n}$.
A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ is a strictly increasing function $\alpha : [n] \hookrightarrow [n']$ satisfying $\sigma = \alpha ^{\ast }( \sigma ' )$ in the set $S_{n}$.
In other words, $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ is the category of elements of the functor $S: \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ (see Variant 5.2.6.2).
Example 3.3.3.13. Let $Q$ be a partially ordered set, and let $\operatorname{N}_{\bullet }(Q)$ denote its nerve. By definition, the nondegenerate $n$-simplices of $\operatorname{N}_{\bullet }(Q)$ can be identified with the strictly increasing functions $\sigma : \{ 0 < 1 < \cdots < n \} \rightarrow Q$. The construction $([n], \sigma ) \mapsto \operatorname{im}(\sigma )$ determines an isomorphism from the category of nondegenerate simplices $\operatorname{{\bf \Delta }}_{\operatorname{N}_{\bullet }(Q)}^{\mathrm{nd}}$ to the partially ordered set $\operatorname{Chain}[Q]$ of Notation 3.3.2.1.
Warning 3.3.3.14. Though the category $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ is defined for any simplicial set $X$, it is primarily useful in the case where $X$ is braced (where we can use the description supplied by Example 3.3.3.12).
Exercise 3.3.3.15. Let $X$ be a simplicial set. Show that $X$ is braced if and only if the inclusion functor $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} \hookrightarrow \operatorname{{\bf \Delta }}_{X}$ admits a left adjoint.
Proposition 3.3.3.16. Let $X$ be a braced simplicial set. Then the comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.9 restricts to an isomorphism $\psi _{X}^{\circ }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) \xrightarrow {\sim } \operatorname{Sd}(X)$.
Example 3.3.3.17. Let $Q$ be a partially ordered set. Combining Proposition 3.3.3.16 with Example 3.3.3.13, we obtain a canonical isomorphism $\operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[Q] )$. In the special case $Q = [n]$, this recovers the isomorphism $\operatorname{Sd}( \Delta ^{n} ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ of Example 3.3.3.5.
The proof of Proposition 3.3.3.16 will make use of the following:
Lemma 3.3.3.18. The functor preserves colimits.
\[ \{ \textnormal{Semisimplicial Sets} \} \rightarrow \{ \textnormal{Simplicial Sets} \} \quad \quad S \mapsto \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}) \]
Proof. Let $n$ be a nonnegative integer. For every semisimplicial set $S$, Example 3.3.3.12 allows us to identify $n$-simplices of the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ with the set of pairs $(\tau , \sigma )$, where $\tau $ is a $m$-simplex of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} )$ (given by a diagram of increasing functions $[k_0] \hookrightarrow [k_1] \hookrightarrow \cdots \hookrightarrow [k_ n]$) and $\sigma $ is an element of the set $S_{k_ n}$. It follows that the functor $S \mapsto \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ preserves colimits. Allowing $n$ to vary, we conclude that the functor $S \mapsto \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ preserves colimits. $\square$
Variant 3.3.3.19. The proof of Lemma 3.3.3.18 also shows that the functor preserves colimits. Consequently, the comparison maps $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.9 are uniquely determined by the following properties:
The construction $X \mapsto \psi _{X}$ is functorial: that is, it determines a natural transformation from the functor $X \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} )$ to the subdivision functor $\operatorname{Sd}$.
When $X = \Delta ^ n$ is a standard simplex, $\psi _{X}$ is the nerve of the functor
described in Example 3.3.3.10.
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \]
\[ \operatorname{{\bf \Delta }}_{X} \rightarrow \operatorname{Chain}[n] \quad \quad (\alpha : [m] \rightarrow [n] \mapsto \mathrm{im}(\alpha ) \subseteq [n] ) \]
Proof of Proposition 3.3.3.16. Let $X$ be a braced simplicial set. By virtue of Corollary 3.3.1.8, we can assume that $X = S^{+}$ for some semisimplicial set $S$. Let $\varphi _{S}$ denote the composite map
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}} ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \xrightarrow { \psi _{X} } \operatorname{Sd}(X) = \operatorname{Sd}(S^{+}); \]
we wish to show that $\varphi _{S}$ is an isomorphism. By virtue of Lemma 3.3.3.18, the construction $S \mapsto \varphi _{S}$ commutes with small colimits. Since every functor $S: \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ can be written as a colimit of representable functors (see §
), we may assume without loss of generality that $S$ is the semisimplicial set represented by an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}$; that is, $X = \Delta ^{n}$ is the standard simplex. In this case, the conclusion follows immediately from the concrete description of $\psi _{X}$ given in Example 3.3.3.10.
$\square$
Remark 3.3.3.20 (Functoriality). Let $u: X \rightarrow Y$ be a morphism of braced simplicial sets. Then $u$ induces a morphism between their subdivisions which can be identified with a functor $U: \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} \rightarrow \operatorname{{\bf \Delta }}_{Y}^{\mathrm{nd}}$ (Proposition 1.3.3.1). If $u$ carries nondegenerate simplices of $X$ to nondegenerate simplices of $Y$, then the functor $U$ is easy to describe: it is given on objects by the formula $U( [n], \sigma ) = ( [n], u(\sigma ) )$. More generally, $U$ carries an object $([n], \sigma ) \in \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ to an object $([m], \tau ) \in \operatorname{{\bf \Delta }}_{Y}^{\mathrm{nd}}$, characterized by the requirement that $u(\sigma )$ factors as a composition $\Delta ^{n} \twoheadrightarrow \Delta ^{m} \xrightarrow {\tau } Y$ (see Proposition 1.1.3.8).
\[ \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) \simeq \operatorname{Sd}(X) \xrightarrow { \operatorname{Sd}(u) } \operatorname{Sd}(Y) \simeq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{Y}^{\mathrm{nd}}), \]
Warning 3.3.3.21. In the statement of Proposition 3.3.3.16, the hypothesis that $X$ is braced cannot be omitted. For example, let $X$ be the simplicial set $\Delta ^{2} \coprod _{ \Delta ^{1} } \Delta ^0$ obtained from the standard $2$-simplex by collapsing a single edge, which we depict informally by the diagram Then the subdivision of $X$ is the $2$-dimensional simplicial set depicted in the diagram This simplicial set cannot arise as the nerve of a category, because it contains a nondegenerate $2$-simplex $\sigma $ for which $d^{2}_2(\sigma )$ is degenerate.
\[ \xymatrix@R =50pt@C=50pt{ & & \bullet \ar [ddddrr] & & \\ & & & & \\ & & & & \\ & & & & \\ \bullet \ar@ {=}[uuuurr] \ar [rrrr] & & & & \bullet .} \]
\[ \xymatrix@R =50pt@C=50pt{ & & \bullet \ar@ {=}[ddl] \ar [ddr] \ar [ddd] & & \\ & & & & \\ & \bullet \ar [dr] & & \bullet \ar [dl] & \\ & & \bullet & & \\ \bullet \ar [urr] \ar [rr] \ar@ {=}[uur] & & \bullet \ar [u] & & \bullet . \ar [ll] \ar [ull] \ar [uul] } \]