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Lemma 1.2.3.13. 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:
- $(1)$
Suppose we are given a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d] & Y \ar [d] \\ X' \ar [r] & Y', } \]
where $f$ is a monomorphism. If $X$, $Y$, and $X'$ belong to $\operatorname{\mathcal{U}}$, then $Y'$ belongs to $\operatorname{\mathcal{U}}$.
- $(2)$
Suppose we are given a sequence of monomorphisms of simplicial sets
\[ X(0) \hookrightarrow X(1) \hookrightarrow X(2) \hookrightarrow X(3) \hookrightarrow \cdots \]
If each $X(m)$ belongs to $\operatorname{\mathcal{U}}$, then the sequentual colimit $\varinjlim _{m} X(m)$ belongs to $\operatorname{\mathcal{U}}$.
- $(3)$
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.
Let $S$ be a simplicial set; we wish to show that $S$ belongs to $\operatorname{\mathcal{U}}$. By virtue of Remark 1.1.4.4, we can identify $S$ with the colimit $\varinjlim _{n} \operatorname{sk}_{n}( S )$. By virtue of $(2)$, it will suffice to show that each skeleton $\operatorname{sk}_{n}(S)$ belongs to $\operatorname{\mathcal{U}}$. We may therefore assume without loss of generality that $S$ has dimension $\leq n$, for some integer $n$. We proceed by induction on $n$. In the case $n=-1$, the simplicial set $S$ is empty and the desired result is a special case of $(3)$. To carry out the inductive step, we invoke Proposition 1.1.4.12 to choose a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in C }{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in C }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( S ) \ar [r] & S, } \]
where $C$ is the collection of nondegenerate $n$-simplices of $S$. By virtue of assumption $(1)$, it will suffice to show that the simplicial sets $\operatorname{sk}_{n-1}( S )$, $\underset { \sigma \in C}{\coprod } \operatorname{\partial \Delta }^{n}$, and $ \underset { \sigma \in C }{\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 $(3)$.
$\square$