$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Lemma 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:


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


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 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


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, 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 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

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, 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 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, 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 $(c)$. $\square$