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5.3.4 The Category of Simplices

Our goal in this section is to show that, up to equivalence, every $\infty $-category can be realized as a localization $\operatorname{\mathcal{C}}[W^{-1}]$, where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. More precisely, we have the following:

Proposition 5.3.4.1. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then there exists an ordinary category $\operatorname{\mathcal{C}}$, a collection $W$ of morphisms in $\operatorname{\mathcal{C}}$, and a functor $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with respect to $W$.

We will deduce Proposition 5.3.4.1 from a more precise assertion (Theorem 5.3.4.4), which gives an explicit construction of the category $\operatorname{\mathcal{C}}$ and the class of morphisms $W$.

Construction 5.3.4.2 (The Last Vertex Map). Let $S$ be a simplicial set, let $\operatorname{{\bf \Delta }}_{S}$ denote the category of simplices of $S$ (Construction 1.1.8.19), and let $\tau $ be a $k$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S})$, which we identify with a commutative diagram of simplicial sets

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

We let $\lambda ^{+}_{S}(\tau ): \Delta ^{k} \rightarrow S$ denote the map 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 composite map $\Delta ^{n_ i} \rightarrow \Delta ^{n_{i+1}} \rightarrow \cdots \rightarrow \Delta ^{n_ k}$. The construction $\tau \mapsto \lambda ^{+}_{S}(\tau )$ determines a morphism of simplicial sets $\lambda ^{+}_{S}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$, which we will refer to as the last vertex map. By construction, it carries each object $([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S}$ to the vertex $\sigma (n) \in S$.

Warning 5.3.4.3. Let $S$ be a simplicial set. We have now assigned two different meanings to the term “last vertex map”:

  • The last vertex map $\lambda _{S}$ of Construction 3.3.4.3, which is a morphism of simplicial sets from the subdivision $\operatorname{Sd}(S)$ to $S$.

  • The last vertex map $\lambda _{S}^+$ of Construction 5.3.4.2, which is a morphism of simplicial sets from the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ to $S$.

These meanings are closely related. If the simplicial set $S$ is braced (Definition 3.3.1.1), then we can identify the subdivision $\operatorname{Sd}(S)$ with the nerve of the full subcategory $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \subseteq \operatorname{{\bf \Delta }}_{S}$ spanned by the nondegenerate simplices of $S$ (Proposition 3.3.3.15). Under this identification, the last vertex map $\lambda _{S}$ of Construction 3.3.4.3 is given by the composition

\[ \operatorname{Sd}(S) \simeq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}^{\mathrm{nd} }) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}) \xrightarrow { \lambda ^{+}_{S} } S, \]

where $\lambda ^{+}_{S}$ is the last vertex map of Construction 5.3.4.2 (Example 3.3.4.7).

Proposition 5.3.4.1 is an immediate consequence of the following:

Theorem 5.3.4.4. Let $S$ be a simplicial set, and let $W_{S}$ be the collection of all morphisms $([n], \sigma ) \rightarrow ([n'], \sigma ')$ in the category of simplices $\operatorname{{\bf \Delta }}_{S}$ for which the underlying map of linearly ordered sets $\alpha : [n] \rightarrow [n']$ satisfies $\alpha (n) = n'$. Then the last vertex map $\lambda _{S}^{+}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$.

The rest of this section is devoted to the proof of Theorem 5.3.4.4. We begin by treating the situation where $S = \Delta ^ n$ is a standard simplex. In this case, we will use the following general observation:

Proposition 5.3.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Suppose that there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(a)$

The functor $F$ carries each element of $W$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

$(b)$

The composition $F \circ G$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$).

$(c)$

There exists a natural transformation $u: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the morphism $u_{C}: C \rightarrow (G \circ F)(C)$ belongs to $W$.

Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

Proof. It follows from $(a)$ that, for every $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a map

\[ \theta _{\operatorname{\mathcal{E}}}: \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1} ], \operatorname{\mathcal{E}})^{\simeq } ) \]

By virtue of Proposition 5.3.1.12, it will suffice to show that $\theta _{\operatorname{\mathcal{E}}}$ is bijective for every $\infty $-category $\operatorname{\mathcal{E}}$. Let $\rho : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1} ], \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )$ be the map given by precomposition with $G$. It follows from $(b)$ that the composition $\rho \circ \theta _{\operatorname{\mathcal{E}}}$ is the identity. We will complete the proof by showing that $\theta _{\operatorname{\mathcal{E}}} \circ \rho $ is also the identity. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. By virtue of Theorem 4.4.4.4, $H$ carries the natural transformation $u$ to an isomorphism $H = H \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow H \circ G \circ F$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, so that the isomorphism class of $H$ is fixed by the composition $\theta _{\operatorname{\mathcal{E}}} \circ \rho $. $\square$

Corollary 5.3.4.6. For each nonnegative integer $n$, the last vertex map

\[ \lambda ^{+}_{\Delta ^ n}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \Delta ^{n} } ) \rightarrow \Delta ^ n \]

exhibits $\Delta ^ n$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\Delta ^{n} } )$ with respect to the collection of morphisms $W_{\Delta ^ n}$ appearing in the statement of Theorem 5.3.4.4.

Proof. Note that we can identify $\lambda _{\Delta ^ n}^{+}$ with a functor of ordinary categories $\operatorname{{\bf \Delta }}_{ \Delta ^ n } \rightarrow [n]$, which carries each simplex $\sigma : \Delta ^ m \rightarrow \Delta ^ n$ to its last vertex $\sigma (m)$. This functor has a right adjoint $G: [n] \rightarrow \operatorname{{\bf \Delta }}_{\Delta ^ n}$, which carries each element $m \in [n]$ to the map $\sigma : \Delta ^ m \rightarrow \Delta ^ n$ which is the identity on vertices. We observe that the composition $\lambda _{\Delta ^ n}^{+} \circ G$ is equal to the identity, and the unit transformation $u: \operatorname{id}_{\operatorname{{\bf \Delta }}_{\Delta ^ n}} \rightarrow G \circ \lambda _{\Delta ^ n}^{+}$ carries each object of $\operatorname{{\bf \Delta }}_{\Delta ^ n}$ to a morphism which belongs to $W_{ \Delta ^ n}$. Applying Proposition 5.3.4.5, we conclude that $\lambda _{\Delta ^ n}^{+}$ exhibits $\Delta ^ n$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\Delta ^{n} } )$ with respect to $W_{\Delta ^ n}$. $\square$

We will prove Theorem 5.3.4.4 for a general simplicial set $S$ by writing $S$ as a colimit of simplices, for which the desired result holds by virtue of Corollary 5.3.4.6. For this, we need the following observation:

Lemma 5.3.4.7. The functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \]

commutes with the formation of colimits.

Proof. Fix an integer $n \geq 0$; we wish to show that the set-valued functor $S \mapsto \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} )$ commutes with colimits. For every $n$-simplex $\tau $ of $\operatorname{N}_{n}(\operatorname{{\bf \Delta }})$, let us write $\operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau }$ for the inverse image of $\tau $ under the natural map $\operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} ) \rightarrow \operatorname{N}_{n}( \operatorname{{\bf \Delta }})$, so that we have a decomposition

\[ \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S} ) \simeq \coprod _{\tau \in \operatorname{N}_{n}(\operatorname{{\bf \Delta }})} \operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau } \]

depending functorially on $S$. It will therefore suffice to show that, for each $\tau $, the functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad S \mapsto \operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau } \]

commutes with colimits. We now observe that, if $\tau $ corresponds to a diagram $[m_0] \rightarrow [m_1] \rightarrow \cdots \rightarrow [m_ n]$ in the category $\operatorname{{\bf \Delta }}$, then $\operatorname{N}_{n}(\operatorname{{\bf \Delta }}_{S})_{\tau }$ can be identified with the set of $m_ n$-simplices of $S$. $\square$

Proof of Theorem 5.3.4.4. Let $S$ be a simplicial set and let $W_{S}$ be the collection of morphisms in $\operatorname{{\bf \Delta }}_{S}$ appearing in the statement of Theorem 5.3.4.4; we wish to show that the last vertex map $\lambda _{S}^{+}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$. Using Lemma 5.3.4.7, we can realize $\lambda _{S}^{+}$ as a filtered colimit of morphisms $\lambda _{S_{\alpha }}^{+}$ (and $W_{S}$ with a filtered colimit of $W_{S_{\alpha } }$), where $\{ S_{\alpha } \} $ is the collection of all finite simplicial subsets of $S$ (Remark 3.5.1.8). By virtue of Proposition 5.3.3.1, it will suffice to show that each $\lambda _{S_{\alpha }}^{+}$ exhibits $S_{\alpha }$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S_{\alpha } } )$ with respect to $W_{S_{\alpha }}$. We may therefore replace $S$ by $S_{\alpha }$, and thereby reduce to the case where the simplicial set $S$ is finite.

Since $S$ is a finite simplicial set, it has dimension $\leq n$ for some integer $n \geq -1$. We proceed by induction on $n$. If $n = -1$, then both $S$ and $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ are empty, and the desired result is clear. We may therefore assume that $n \geq 0$ and that Theorem 5.3.4.4 holds for all finite simplicial sets of dimension $< n$. We now proceed by induction on the number $m$ of nondegenerate $n$-simplices of $S$. If $m = 0$, then $S$ has dimension $\leq n-1$ and the desired result holds by virtue of our first inductive hypothesis. We may therefore assume that $S$ has at least one nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow S$. Using Proposition 1.1.3.13, we see that there is a pushout diagram 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 $\leq n$ with exactly $(m-1)$-nondegenerate $m$-simplices. It follows from our inductive hypotheses together with Corollary 5.3.4.6 that the simplicial sets $\operatorname{\partial \Delta }^ n$, $\Delta ^ n$, and $S'$ satisfy the conclusion of Theorem 5.3.4.4. Moreover, the last vertex map $\lambda _{S}^{+}$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \operatorname{\partial \Delta }^ n} ) \ar [dr]^{ \lambda _{\operatorname{\partial \Delta }^ n}^{+} } \ar [rr] \ar [dd] & & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ \Delta ^ n} ) \ar [dr]^{ \lambda _{\Delta ^ n}^{+} } \ar [dd] & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S'} ) \ar [dr]^{ \lambda _{S'}^{+} } \ar [rr] & & \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \ar [dr]^{ \lambda _{S}^{+} } & \\ & S' \ar [rr] & & S, } \]

where the front and back faces are pushout squares (Lemma 5.3.4.7) in which the horizontal maps are monomorphisms, hence categorical pushout squares (Example 4.5.3.8). Since $W_{S}$ is the union of the images of $W_{S'}$ and $W_{ \Delta ^ n }$, it follows from Proposition 5.3.3.2 that $\lambda _{S}^{+}$ exhibits $S$ as a localization of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ with respect to $W_{S}$. $\square$

Exercise 5.3.4.8. Let $S$ be a simplicial set and let $\operatorname{Sd}(S)$ denote the subdivision of $S$. Let $W$ be the collection of all edges $w$ of $\operatorname{Sd}(S)$ with the following property: there exists a simplex $\sigma : \Delta ^ n \rightarrow S$ such that $w = \operatorname{Sd}(\sigma )(\overline{w})$, where $\overline{w}$ is an edge of the subdivision $\operatorname{Sd}( \Delta ^ n ) \simeq \operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ corresponding to a pair of subsets $\emptyset \neq P \subseteq Q \subseteq [n]$ satisfying $\mathrm{max}(P) = \mathrm{max}(Q)$. Show that the last vertex map $\lambda _{S}: \operatorname{Sd}(S) \rightarrow S$ of Construction 3.3.4.3 exhibits $S$ as a localization of $\operatorname{Sd}(S)$ with respect to $W$.