Roughly speaking, one can think of the simplicial sets $\Delta ^ n$ of Example 1.1.0.9 as elementary building blocks out of which more complicated simplicial sets can be constructed. In this section, we make this idea more precise by introducing the skeletal filtration of a simplicial set. This filtration allows us to write every simplicial set $S$ as the union of an increasing sequence of simplicial subsets
where each $\operatorname{sk}_{n}( S )$ is obtained from $\operatorname{sk}_{n-1}( S )$ by attaching copies of $\Delta ^ n$ (see Proposition 1.1.4.12 below for a precise statement).
Construction 1.1.4.1. Let $S = S_{\bullet }$ be a simplicial set and let $k$ be an integer. For every integer $n$, we let $\operatorname{sk}_{k}(S)_{n}$ denote the subset of $S_{n}$ consisting of those $n$-simplices $\sigma : \Delta ^ n \rightarrow S$ which satisfy the following condition:
In the category of simplicial sets, $\sigma $ admits a factorizaton
where $m \leq k$.
\[ \Delta ^{n} \rightarrow \Delta ^{m} \xrightarrow { \tau } S \]
It follows immediately from the definitions that the collection of subsets $\{ \operatorname{sk}_{k}(S)_{n} \subseteq S_{n} \} _{n \geq 0}$ is stable under the face and degeneracy operators for the simplicial set $S_{\bullet }$, and therefore defines a simplicial subset $\operatorname{sk}_{k}(S) \subseteq S$. We will refer to $\operatorname{sk}_{k}(S)$ as the $k$-skeleton of $S$.
Example 1.1.4.2. For every simplicial set $S$, the $k$-skeleton $\operatorname{sk}_{k}( S)$ is empty for $k < 0$.
Remark 1.1.4.3. Let $m$ and $n$ be integers with $m \leq n$. Then, for every simplicial set $S$, the $m$-skeleton $\operatorname{sk}_{m}(S)$ is contained in the $n$-skeleton $\operatorname{sk}_{n}(S)$.
Remark 1.1.4.4. Let $S$ be a simplicial set and let $k$ be an integer. If $n \leq k$, then $\operatorname{sk}_{k}( S )$ contains every $n$-simplex of $S$. In particular, the union $\bigcup _{k} \operatorname{sk}_{k}( S )$ is equal to $S$.
Remark 1.1.4.5. Let $S$ be a simplicial set and let $\sigma $ be a nondegenerate $n$-simplex of $S$. Then $\sigma $ is contained in the $k$-skeleton $\operatorname{sk}_{k}(S)$ if and only if $n \leq k$ (see Proposition 1.1.2.10).
Proposition 1.1.4.6. Let $S$ be a simplicial set and let $k$ be an integer. Then:
The simplicial set $\operatorname{sk}_{k}( S )$ has dimension $\leq k$.
For every simplicial set $T$ of dimension $\leq k$, composition with the inclusion map $\operatorname{sk}_{k}( S ) \hookrightarrow S$ induces a bijection
In other words, the image of any map $T \rightarrow S$ is contained in $\operatorname{sk}_{k}( S )$.
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( T, \operatorname{sk}_{k}( S ) ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( T, S). \]
Proof. Assertion $(a)$ follows from Remark 1.1.4.5. To prove $(b)$, suppose that $f: T \rightarrow S$ is a map of simplicial sets, where $T$ has dimension $\leq k$. We wish to show that $f$ carries every $n$-simplex $\sigma $ of $T$ to an $n$-simplex of $\operatorname{sk}_{k}( S )$. Using Proposition 1.1.3.8, we can reduce to the case where $\sigma $ is a nondegenerate $n$-simplex of $T$. In this case, our assumption that $T$ has dimension $\leq k$ guarantees that $n \leq k$, so that $f( \sigma )$ belongs to $\operatorname{sk}_{k}( S)$ by virtue of Remark 1.1.4.4. $\square$
Corollary 1.1.4.7. Let $S$ be a simplicial set. For every integer $k$, the $k$-skeleton $\operatorname{sk}_{k}(S)$ is the largest simplicial subset of $S$ of dimension $\leq k$.
Corollary 1.1.4.8. Let $k$ be an integer, let $S$ be a simplicial set, and let $\operatorname{{\bf \Delta }}_{S, \leq k}$ denote the category of simplices of $S$ having dimension $\leq k$ (see Construction 1.1.3.9). Then the tautological map is a monomorphism, whose image is the $k$-skeleton $\operatorname{sk}_{k}(S) \subseteq S$.
\[ \varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k} } \rightarrow S \]
Proof. By virtue of Remark 1.1.4.4, replacing $S$ by the $k$-skeleton $\operatorname{sk}_{k}(S)$ does not change the category $\operatorname{{\bf \Delta }}_{S, \leq k}$. We may therefore assume without loss of generality that $S$ has dimension $\leq k$, in which case the desired result follows from Proposition 1.1.3.11. $\square$
Corollary 1.1.4.9. For every integer $k$, the skeleton functor $\operatorname{sk}_{k}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ preserves small colimits.
Proof. Let $S: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets; we wish to show that the comparison map
is an isomorphism of simplicial sets. Using Propositions 1.1.4.6 and 1.1.3.11, we see that the source and target of $\theta $ are simplicial sets of dimension $\leq k$. It will therefore suffice to show that $\theta $ induces a bijection on $n$-simplices for $n \leq k$ (Corollary 1.1.3.14), which follows immediately from Remark 1.1.4.4 (and Remark 1.1.0.8). $\square$
Construction 1.1.4.10 (The Boundary of $\Delta ^ n$). Let $n \geq 0$ be an integer and let $\Delta ^{n}$ denote the standard $n$-simplex (Example 1.1.0.9). We let $\operatorname{\partial \Delta }^{n}$ denote the $(n-1)$-skeleton of $\Delta ^{n}$. We will refer to $\operatorname{\partial \Delta }^ n$ as the boundary of $\Delta ^ n$. More explicitly, the simplicial set $(\operatorname{\partial \Delta }^ n): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is defined by the formula
\[ ( \operatorname{\partial \Delta }^{n} )( [m] ) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ): \text{$\alpha $ is not surjective} \} . \]
Example 1.1.4.11. The simplicial set $\operatorname{\partial \Delta }^{0}$ is empty.
Let $S$ be a simplicial set. For each $k \geq 0$, we let $S_{k}^{\mathrm{nd}}$ denote the collection of all nondegenerate $k$-simplices of $S$. Every element $\sigma \in S_{k}^{\mathrm{nd}}$ determines a map of simplicial sets $\Delta ^{k} \rightarrow \operatorname{sk}_{k}( S )$. Since the boundary $\operatorname{\partial \Delta }^ k \subseteq \Delta ^{k}$ has dimension $\leq k-1$, this map carries $\operatorname{\partial \Delta }^{k}$ into the $(k-1)$-skeleton $\operatorname{sk}_{k-1}( S)$ (Proposition 1.1.4.6).
Proposition 1.1.4.12. Let $S$ be a simplicial set and let $k \geq 0$. Then the construction outlined above determines a pushout square in the category $\operatorname{Set_{\Delta }}$ of simplicial sets.
\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in S_{k}^{\mathrm{nd}} }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { \sigma \in S_{k}^{\mathrm{nd}} }{\coprod } \Delta ^{k} \ar [d] \\ \operatorname{sk}_{k-1}( S ) \ar [r] & \operatorname{sk}_{k}( S ) } \]
Proof. Unwinding the definitions, we must prove the following:
Let $\tau $ be an $n$-simplex of $\operatorname{sk}_{k}( S )$ which is not contained in $\operatorname{sk}_{k-1}(S)$. Then $\tau $ factors uniquely as a composition
where $\sigma $ is a nondegenerate simplex of $S$ and $\alpha $ does not factor through the boundary $\operatorname{\partial \Delta }^{k}$ (in other words, $\alpha $ is surjective on vertices).
Proposition 1.1.3.8 implies that any $n$-simplex of $S$ admits a unique factorization $\Delta ^{n} \xrightarrow {\alpha } \Delta ^{m} \xrightarrow {\sigma } S$, where $\alpha $ is surjective on vertices and $\sigma $ is nondegenerate. Our assumption that $\tau $ belongs to the $\operatorname{sk}_{k}(S)$ guarantees that $m \leq k$, and our assumption that $\tau $ does not belong to $\operatorname{sk}_{k-1}( S )$ guarantees that $m \geq k$. $\square$
We close this section by analyzing the simplicial sets $\operatorname{\partial \Delta }^ n$ of Construction 1.1.4.10 in a bit more detail. Note that, for every pair of integers $0 \leq k \leq n$, the morphism $\delta ^{k}_{n}: \Delta ^{n-1} \rightarrow \Delta ^ n$ of Construction 1.1.1.4 factors through the boundary $\operatorname{\partial \Delta }^{n}$.
Proposition 1.1.4.13. Let $n$ be a positive integer. For every simplicial set $S_{\bullet }$, the map is an injection, whose image consists of those tuples of $( \sigma _0, \sigma _1, \cdots , \sigma _ n )$ of $(n-1)$-simplices of $S$ which satisfy the identity $d^{n-1}_ i(\sigma _ j) = d^{n-1}_{j-1}( \sigma _{i})$ for $0 \leq i < j \leq n$.
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n}, S_{\bullet } ) \rightarrow (S_{n-1})^{n+1} \quad \quad f \mapsto \{ f \circ \delta ^{k}_{n} \} _{0 \leq k \leq n} \]
Example 1.1.4.14. When $n = 1$, Proposition 1.1.4.13 asserts that we can identify maps $\operatorname{\partial \Delta }^1 \rightarrow S$ with ordered pairs $(s,t)$ of vertices of $S$. Equivalently, the boundary $\operatorname{\partial \Delta }^1$ can be identified with the coproduct of $\{ 0\} $ and $\{ 1\} $ (which we regard as simplicial subsets of $\Delta ^1$ as in Example 1.1.0.15).
Example 1.1.4.15. When $n=2$, Proposition 1.1.4.13 asserts that morphisms of simplicial sets $\operatorname{\partial \Delta }^2 \rightarrow S$ can be identified with ordered triples $(g, h, f)$ of edges of $S$ having the property that $f$ and $h$ have the same source vertex $x \in S$, $g$ and $h$ have the same target vertex $z \in S$, and the target $y$ of $f$ coincides with the source of $g$; these relationships are summarized visually in the diagram
\[ \xymatrix { & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]
Proof of Proposition 1.1.4.13. Let $w: \coprod _{0 \leq k \leq n} \Delta ^{n-1} \rightarrow \operatorname{\partial \Delta }^{n}$ be the map given on the $k$th summand by $\delta ^{k}_{n}$. To prove the first assertion of Proposition 1.1.4.13, we must show that $w$ is an epimorphism of simplicial sets: that is, it is surjective on $m$-simplices for each $m \geq 0$. In fact, we can be a bit more precise. Let $\alpha $ be an $m$-simplex of $\Delta ^{n}$, which we identify with a nondecreasing function from $[m]$ to $[n]$. Then $\alpha $ belongs to the boundary $\operatorname{\partial \Delta }^{n}$ if and only if it is not surjective: that is, if and only if there exists some integer $0 \leq i \leq n$ such that $\alpha $ factors through $[n] \setminus \{ i\} $. In this case, there is a unique $m$-simplex $\beta _{i}$ which belongs to the $i$th summand of $\coprod _{0 \leq k \leq n} \Delta ^{n-1}$ and satisfies $w( \beta _ i ) = \alpha $.
For every integer $0 \leq k \leq n$, let $u_{k}: \coprod _{ 0 \leq i < k } \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ be the map given on the $i$th summand by $\delta ^{i}_{n-1}$, and let $v_{k}: \coprod _{ k < j \leq n} \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ by the map given on the $j$th summand by $\delta ^{j-1}_{n-1}$. Passing to the coproduct over $k$ and reindexing, we obtain a pair of maps
Let $\operatorname{Coeq}(u,v)_{\bullet }$ denote the coequalizer of $u$ and $v$ in the category of simplicial sets. The morphism $w$ satisfies $w \circ u = w \circ v$ (see Remark 1.1.1.7), and therefore factors uniquely through a map $\overline{w}: \operatorname{Coeq}(u,v)_{\bullet } \rightarrow \operatorname{\partial \Delta }^{n}$. Proposition 1.1.4.13 asserts that $\overline{w}$ is an isomorphism of simplicial sets: that is, for every integer $m \geq 0$, it induces a bijection from $\operatorname{Coeq}(u,v)_{m}$ to the set of $m$-simplices of $\operatorname{\partial \Delta }^{n}$. The surjectivity of this map was established above. To prove injectivity, it will suffice to observe that if $\alpha : [m] \rightarrow [n]$ is as above and we are given two elements $i,j \in [n]$ which do not belong to the image of $\alpha $, then $\beta _{i}$ and $\beta _{j}$ have the same image in $\operatorname{Coeq}(u,v)_{\bullet }$. If $i = j$, this is automatic; we may therefore assume without loss of generality that $i < j$. In this case, the desired result follows from the observation that we can write $\beta _ j = u(\gamma )$ and $\beta _ i = v(\gamma )$, where $\gamma $ is the $m$-simplex of the $(i,j)$th summand of $\coprod _{ 0 \leq i < j \leq n} \Delta ^{n-2}$ corresponding to the nondecreasing function $[m] \xrightarrow {\alpha } [n] \setminus \{ i < j \} \simeq [n-2]$. $\square$