Lemma 9.2.2.17. Let $B$ be a linearly ordered set and let $A \subseteq B$ be a subset which satisfies the following condition:
- $(\ast )$
For every element $b \in B$, the set $\{ a \in A: a \leq b \} $ has a largest element $b_{-}$, and the set $\{ a \in A: b \leq a \} $ has a smallest element $b_{+}$.
Let $K(A,B) \subseteq \operatorname{N}_{\bullet }(B)$ be the simplicial subset whose $n$-simplices are given by tuples $(b_0 \leq b_1 \leq b_2 \leq \cdots \leq b_ n)$ which satisfy one of the following conditions:
Each of the elements $b_{i}$ belongs to $A$.
For every element $a \in A$, either $a \leq b_0$ or $a \geq b_ n$.
Then the inclusion map $\iota : K(A,B) \hookrightarrow \operatorname{N}_{\bullet }( B )$ is a categorical equivalence of simplicial sets.
Proof.
Note that we can identify $K(A,B)$ with the (filtered) colimit of the simplicial subsets $K(A, B')$, where $B'$ ranges over the collection of all subsets of $B$ which are obtained from $A$ by adjoining finitely many elements. Since the collection of categorical equivalences is stable under the formation of filtered colimits (Corollary 4.5.7.2), it will suffice to prove Lemma 9.2.2.17 in the special case where $B \setminus A$ is finite.
Let $A_0 \subseteq A$ be the collection of elements which have the form $b_{-}$ or $b_{+}$, where $b$ is an element of $B \setminus A$. Note that, if $A' \subseteq A$ is a subset which contains $A_0$ and we set $B' = A' \cup (B \setminus A)$, then the pair $(A', B')$ also satisfies condition $(\ast )$. Moreover, we have $K(A',B') = K(A,B) \cap \operatorname{N}_{\bullet }(B')$. It follows that $K(A,B)$ can be written as a filtered colimit of simplicial subsets $K(A',B')$, where $A'$ ranges over finite subsets of $A$ which contain $A_0$. Applying Corollary 4.5.7.2 again, we are reduced to proving Lemma 9.2.2.17 under the additional assumption that $A$ is finite. We will also assume that $A$ is nonempty (otherwise, $B$ is empty and therefore is nothing to prove).
If $B = \emptyset $, there is nothing to prove. We may therefore assume without loss of generality that $B = [n] = \{ 0 < 1 < \cdots n \} $ for some nonnegative integer $n$, so that $\operatorname{N}_{\bullet }(B)$ can be identified with the standard $n$-simplex $\Delta ^ n$. Note that the simplicial subset $K(A,B) \subseteq \Delta ^ n$ contains the spine $\operatorname{Spine}[n]$ of Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7), and therefore a categorical equivalence. It will therefore suffice to show that the inclusion map $\operatorname{Spine}[n] \hookrightarrow K(A,B)$ is also a categorical equivalence. In fact, we will show that it is inner anodyne.
Write $A = \{ a_0 < a_1 < \cdots < a_ m \} $, where $a_0 = 0$ and $a_ m = n$. Then $\operatorname{N}_{\bullet }(A)$ is the image of a nondegenerate $m$-simplex $\sigma : \Delta ^{m} \rightarrow \Delta ^{n}$, given by $\sigma (i) = a_ i$. Let $K' \subseteq \Delta ^ n$ denote the simplicial subset consisting of simplices which satisfy condition $(1)$: more concretely, $K'$ is the union of the images of nondegenerate simplices
\[ \tau _{i}: \Delta ^{ a_{i} - a_{i-1} } \rightarrow \Delta ^{n} \quad \quad k \mapsto k + a_{i-1}. \]
Note that the inverse image $\sigma ^{-1}( K' )$ identifies with the spine $\operatorname{Spine}[m]$, so that we have a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Spine}[m] \ar [r] \ar [d] & \Delta ^{m} \ar [d]^{\sigma } \\ K' \ar [r] & K(A,B). } \]
Since the inclusion map $\operatorname{Spine}[m] \hookrightarrow \Delta ^{m}$ is inner anodyne (Example 1.5.7.7), it follows that the inclusion $K' \hookrightarrow K(A,B)$ is also inner anodyne. We are therefore reduced to showing that the inclusion map $\operatorname{Spine}[n] \hookrightarrow K'$ is inner anodyne. This follows from the observation that we also have a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{1 \leq i \leq m} \operatorname{Spine}[ a_{i} - a_{i-1} ] \ar [r] \ar [d] & \operatorname{Spine}[n] \ar [d] \\ \coprod _{1 \leq i \leq m} \Delta ^{a_{i} - a_{i-1} } \ar [r]^-{ \{ \tau _ i \} } & K', } \]
where the left vertical map is inner anodyne by virtue of Example 1.5.7.7.
$\square$