Kerodon

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Lemma 4.5.8.10. Let $Y$ be a simplicial set, and suppose that the comparison map $c_{\Delta ^1, Y}: \Delta ^1 \diamond Y \rightarrow \Delta ^1 \star Y$ is a categorical equivalence. Then, for every simplicial set $X$, the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ is a categorical equivalence.

Proof. Throughout the proof, we regard the simplicial set $Y$ as fixed. Let us say that a simplicial set $X$ is good if $c_{X,Y}$ is a categorical equivalence. We begin with some elementary observations:

$(a)$

The collection of good simplicial sets is closed under the formation of filtered colimits (since the collection of categorical equivalences is closed under filtered colimits, by virtue of Corollary 4.5.7.2).

$(b)$

Suppose we are given a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d] & X(0) \ar [d] \\ X(1) \ar [r] & X(01), }$

where $f$ is a monomorphism. If $X$, $X(0)$, and $X(1)$ are good, then $X(01)$ is good. This follows by applying Proposition 4.5.4.9 to the commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [dr]^{ c_{X,Y} } \ar [rr] \ar [dd] & & X(0) \diamond Y \ar [dr]^{ c_{X(0),Y} } \ar [dd] & \\ & X \star Y \ar [rr] \ar [dd] & & X(0) \star Y \ar [dd] \\ X(1) \diamond Y \ar [dr]^{ c_{X(1),Y}} \ar [rr] & & X(01) \diamond Y \ar [dr]^{ c_{X(01),Y} } & \\ & X(1) \star Y \ar [rr] & & X(01) \star Y, }$

noting that the front and back squares are categorical pushouts by virtue of Example 4.5.4.12.

$(c)$

Let $f: X \rightarrow X'$ be an inner anodyne morphism of simplicial sets. Then $X$ is good if and only if $X'$ is good. To prove this, we observe that there is a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [r]^-{ c_{X,Y} } \ar [d]^{f \diamond \operatorname{id}_ Y} & X \star Y \ar [d]^{f \star \operatorname{id}_{Y}} \\ X' \diamond Y \ar [r]^-{c_{Y} } & X' \star Y. }$

By the two-out-of-three property (Remark 4.5.3.5), it will suffice to show that the morphisms $f \diamond \operatorname{id}_{Y}$ and $f \star \operatorname{id}_{Y}$ are categorical equivalences. In the first case, this follows from Remark 4.5.8.7. For the second, we observe that $f \star \operatorname{id}_{Y}$ is actually inner anodyne, since it factors as a composition

$X \star Y \xrightarrow {u} X' \coprod _{X} (X \star Y ) \xrightarrow {v} X' \star Y,$

where $u$ is a pushout of $f$ (hence inner anodyne because $f$ is inner anodyne) and $v$ is inner anodyne by virtue of Proposition 4.3.6.4.

We wish to show that if the $1$-simplex $\Delta ^1$ is good, then every simplicial set $X$ is good. Writing $X$ as the filtered colimit of its finite simplicial subsets (Remark 3.5.1.8), we can use $(a)$ to reduce to the case where $X$ is finite. We now proceed by induction on the dimension of $X$. If $X = \emptyset$, then $c_{X,Y}$ is an isomorphism (Example 4.5.8.4). Otherwise, the simplicial set $X$ has dimension $n \geq 0$. We now proceed by induction on the number of nondegenerate $n$-simplices of $X$. Using Proposition 1.1.3.13, we can choose a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \Delta ^{n} \ar [d] \\ X' \ar [r] & X }$

where $X' \subseteq X$ is a simplicial subset having one fewer nondegenerate $n$-simplex. It then follows from our inductive hypothesis that $\operatorname{\partial \Delta }^ n$ and $X'$ are good. By virtue of $(b)$, it will suffice to show that $\Delta ^ n$ is good. This holds for $n=1$ by assumption, and also for $n=0$ because $\Delta ^{0}$ is a retract of $\Delta ^{1}$. We may therefore assume that $n \geq 2$, so that the horn inclusion $\Lambda ^{n}_{1} \hookrightarrow \Delta ^ n$ is inner anodyne. Our inductive hypothesis guarantees that $\Lambda ^{n}_{1}$ is good, so that $\Delta ^ n$ is good by virtue of $(c)$. $\square$