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Proposition 3.4.5.4. For every simplicial set $X$, the counit map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence.

Proof of Proposition 3.4.5.4. We proceed as in the proof of Proposition 3.3.4.8. For every simplicial set $X$, the counit map $v_{X}: X^{+} \rightarrow X$ can be realized as a filtered colimit of counit maps $\{ v_{ \operatorname{sk}_{n}(X)}: \operatorname{sk}_{n}(X)^{+} \rightarrow \operatorname{sk}_{n}(X) \} _{n \geq 0}$. Since the collection of weak homotopy equivalences is closed under the formation of filtered colimits (Proposition 3.2.8.3), it will suffice to show that each of the maps $v_{ \operatorname{sk}_{n}(X)}$ is a weak homotopy equivalence. We may therefore replace $X$ by $\operatorname{sk}_{n}(X)$, and thereby reduce to the case where $X$ is $n$-skeletal for some nonnegative integer $n \geq 0$. We now proceed by induction on $n$.

Let $Y = \operatorname{sk}_{n-1}(X)$ be the $(n-1)$-skeleton of $X$. Let $S$ denote the collection of nondegenerate $n$-simplices of $X$, let $X' = \coprod _{\sigma \in S} \Delta ^{n}$ denote their coproduct, and let $Y' = \coprod _{\sigma \in S} \operatorname{\partial \Delta }^{n}$ denote the boundary of $X'$. Proposition 1.1.3.13 then supplies a pushout diagram of simplicial sets

3.62
\begin{equation} \label{equation:pushout-aux-1} \begin{gathered} \xymatrix@R =50pt@C=50pt{ Y' \ar [r] \ar [d] & X' \ar [d] \\ Y \ar [r] & X. } \end{gathered} \end{equation}

Note that both (3.62) and the induced diagram

\[ \xymatrix@R =50pt@C=50pt{ Y'^{+} \ar [r] \ar [d] & X'^{+} \ar [d] \\ Y^{+} \ar [r] & X^{+} } \]

are homotopy pushout squares (this is a special case of Example 3.4.2.12, since the maps $Y' \hookrightarrow X'$ and $Y'^{+} \hookrightarrow X'^{+}$ are monomorphisms). Moreover, our inductive hypothesis guarantees that the maps $v_{Y}: Y^{+} \rightarrow Y$ and $v_{Y'}: Y'^{+} \rightarrow Y'$ are weak homotopy equivalences. Applying Proposition 3.4.2.9 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ Y'^{+} \ar [rr] \ar [dd] \ar [dr]^{ v_{Y'} } & & Y^{+} \ar [dr]^{v_{Y}} \ar [dd] & \\ & Y' \ar [rr] \ar [dd] & & Y \ar [dd] \\ X'^{+} \ar [rr] \ar [dr]^{v_{X'} } & & X^{+} \ar [dr]^{v_{X}} & \\ & X' \ar [rr] & & X, } \]

we are reduced to proving that $v_{X'}$ is a weak homotopy equivalence. Using Remark 3.1.6.20, we can reduce further to the problem of showing that the map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence in the special case $X = \Delta ^{n}$, which follows from Lemma 3.4.5.9. $\square$