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

Lemma Let $\operatorname{\mathcal{U}}$ be a full subcategory of the category $\operatorname{Set_{\Delta }}$ of simplicial sets. Suppose that $\operatorname{\mathcal{U}}$ satisfies the following three conditions:


Suppose we are given a pushout diagram of simplicial sets

\[ \xymatrix { X_{\bullet } \ar [r]^{f} \ar [d] & Y_{\bullet } \ar [d] \\ X'_{\bullet } \ar [r] & Y'_{\bullet }, } \]

where $f$ is a monomorphism. If $X_{\bullet }$, $Y_{\bullet }$, and $X'_{\bullet }$ belong to $\operatorname{\mathcal{U}}$, then $Y'_{\bullet }$ belongs to $\operatorname{\mathcal{U}}$.


Suppose we are given a sequence of monomorphisms of simplicial sets

\[ X(0)_{\bullet } \hookrightarrow X(1)_{\bullet } \hookrightarrow X(2)_{\bullet } \hookrightarrow X(3)_{\bullet } \hookrightarrow \cdots \]

If each $X(m)_{\bullet }$ belongs to $\operatorname{\mathcal{U}}$, then the sequentual colimit $\varinjlim _{m} X(m)_{\bullet }$ belongs to $\operatorname{\mathcal{U}}$.


For each $n \geq 0$ and every set $I$, the coproduct $\coprod _{i \in I} \Delta ^{n}$ belongs to $\operatorname{\mathcal{U}}$.

Then every simplicial set belongs to $\operatorname{\mathcal{U}}$.

Proof. Set $S_{\bullet }$ be a simplicial set; we wish to show that $S_{\bullet }$ belongs to $\operatorname{\mathcal{U}}$. By virtue of Remark, we can identify $S_{\bullet }$ with the colimit $\varinjlim _{n} \operatorname{sk}_{n}( S_{\bullet } )$. By virtue of $(2)$, it will suffice to show that each skeleton $\operatorname{sk}_{n}(S_{\bullet })$ belongs to $\operatorname{\mathcal{U}}$. We may therefore assume without loss of generality that $S_{\bullet }$ has dimension $\leq n$, for some integer $n$. We proceed by induction on $n$. In the case $n=-1$, the simplicial set $S_{\bullet }$ is empty, and the desired result is a special case of $(3)$. To carry out the inductive step, we invoke Proposition to choose a pushout diagram

\[ \xymatrix { \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S_{\bullet } ) \ar [r] & S_{\bullet } . } \]

By virtue of assumption $(1)$, it will suffice to show that the simplicial sets $\operatorname{sk}_{n-1}( S_{\bullet } )$, $\underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{n}$, and $ \underset { \sigma \in S_{n}^{\mathrm{nd}} }{\coprod } \Delta ^{n}$ belong to $\operatorname{\mathcal{U}}$. In the first two cases, this follows from our inductive hypothesis. In the third, it follows from assumption $(3)$. $\square$