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1.1.2 Simplices and Horns

We now consider some elementary examples of simplicial sets.

Construction 1.1.2.1 (The Standard Simplex). Let $n \geq 0$ be an integer. We let $\Delta ^{n}$ denote the simplicial set given by the construction

\[ ([m] \in \operatorname{{\bf \Delta }}) \mapsto \operatorname{Hom}_{ \operatorname{{\bf \Delta }}}( [m], [n] ). \]

We will refer to $\Delta ^{n}$ as the standard $n$-simplex. By convention, we extend this construction to the case $n = -1$ by setting $\Delta ^{-1} = \emptyset $.

Example 1.1.2.2. The standard $0$-simplex $\Delta ^{0}$ is a final object of the category of simplicial sets: that is, it carries each $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to a set having a single element.

Remark 1.1.2.3. For each $n \geq 0$, the standard $n$-simplex $\Delta ^ n$ is characterized by the following universal property: for every simplicial set $X_{\bullet }$, Yoneda's lemma supplies a bijection

\[ \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^ n, X_{\bullet } ) \simeq X_{n}. \]

We will often invoke this bijection implicitly to identify $n$-simplices of $X_{\bullet }$ with maps of simplicial sets $\sigma : \Delta ^ n \rightarrow X_{\bullet }$.

Remark 1.1.2.4. Let $S_{\bullet }$ be a simplicial set. Suppose that, for every integer $n \geq 0$, we are given a subset $T_{n} \subseteq S_{n}$, and that the face and degeneracy maps

\[ d_{i}: S_{n} \rightarrow S_{n-1} \quad \quad s_{i}: S_{n} \rightarrow S_{n+1} \]

carry $T_{n}$ into $T_{n-1}$ and $T_{n+1}$, respectively. Then the collection $\{ T_{n} \} _{n \geq 0}$ inherits the structure of a simplicial set $T_{\bullet }$. In this case, we will say that $T_{\bullet }$ is a simplicial subset of $S_{\bullet }$ and write $T_{\bullet } \subseteq S_{\bullet }$.

Example 1.1.2.5. Let $S_{\bullet }$ be a simplicial set and let $v$ be a vertex of $S_{\bullet }$. Then $v$ can be identified with a map of simplicial sets $\Delta ^0 \rightarrow S_{\bullet }$. This map is automatically a monomorphism (note that $\Delta ^0$ has only a single $n$-simplex for every $n \geq 0$), whose image is a simplicial subset of $S_{\bullet }$. It will often be convenient to denote this simplicial subset by $\{ v \} $. For example, we can identify vertices of the standard $n$-simplex $\Delta ^ n$ with integers $i$ satisfying $0 \leq i \leq n$; every such integer $i$ determines a simplicial subset $\{ i \} \subseteq \Delta ^ n$ (whose $k$-simplices are the constant maps $[k] \rightarrow [n]$ taking the value $i$).

It will be useful to consider some other simplicial subsets of the standard $n$-simplex.

Construction 1.1.2.6 (The Boundary of $\Delta ^ n$). Let $n \geq 0$ be an integer. We define a simplicial set $(\operatorname{\partial \Delta }^ n): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ by the formula

\[ ( \operatorname{\partial \Delta }^{n} )( [m] ) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ): \text{$\alpha $ is not surjective} \} . \]

Note that we can regard $\operatorname{\partial \Delta }^ n$ as a simplicial subset of the standard $n$-simplex $\Delta ^ n$ of Construction 1.1.2.1. We will refer to $\operatorname{\partial \Delta }^ n$ as the boundary of $\Delta ^ n$.

Example 1.1.2.7. The simplicial set $\operatorname{\partial \Delta }^{0}$ is empty.

Exercise 1.1.2.8. Let $n \geq 0$ be an integer. For $0 \leq j \leq n$, the map $\delta ^{j}: [n-1] \rightarrow [n]$ of Notation 1.1.1.6 determines a map of simplicial sets $\Delta ^{n-1} \rightarrow \Delta ^{n}$ which factors through the simplicial subset $\operatorname{\partial \Delta }^ n \subseteq \Delta ^ n$. We therefore obtain a map of simplicial sets $\Delta ^{n-1} \rightarrow \operatorname{\partial \Delta }^{n}$, which we will also denote by $\delta ^{j}$. Show that, for any simplicial set $S_{\bullet }$, the construction

\[ ( f: \operatorname{\partial \Delta }^{n} \rightarrow S_{\bullet } ) \mapsto \{ f \circ \delta ^{j} \} _{0 \leq j \leq n} \]

determines an injective map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n}, S_{\bullet } ) \rightarrow \prod _{ j \in [n]} S_{n-1}, \]

whose image is the collection of sequences of $(n-1)$-simplices $(\sigma _0, \sigma _1, \ldots , \sigma _ n)$ satisfying the identities $d_ j(\sigma _ k) = d_{k-1}( \sigma _{j})$ for $0 \leq j < k \leq n$.

Construction 1.1.2.9 (The Horn $\Lambda ^{n}_{i}$). Suppose we are given a pair of integers $0 \leq i \leq n$. We define a simplicial set $\Lambda ^{n}_{i}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ by the formula

\[ ( \Lambda ^{n}_{i} )([m]) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}([m], [n]): [n] \nsubseteq \alpha ([m] ) \cup \{ i \} \} . \]

We regard $\Lambda ^{n}_{i}$ as a simplicial subset of the boundary $\operatorname{\partial \Delta }^{n} \subseteq \Delta ^ n$. We will refer to $\Lambda ^{n}_{i}$ as the $i$th horn in $\Delta ^{n}$. We will say that $\Lambda ^{n}_{i}$ is an inner horn if $0 < i < n$, and an outer horn if $i = 0$ or $i = n$.

Remark 1.1.2.10. Roughly speaking, one can think of the horn $\Lambda ^{n}_{i}$ as obtained from the $n$-simplex $\Delta ^ n$ by removing its interior together with the face opposite its $i$th vertex (see Example 1.1.6.13).

Example 1.1.2.11. The horns contained in $\Delta ^2$ are depicted in the following diagram:

\[ \xymatrix { & \{ 1\} \ar@ {-->}[ddr] & & & \{ 1\} \ar [ddr] & & & \{ 1\} \ar [ddr] & \\ & \Lambda ^{2}_{0} & & & \Lambda ^{2}_{1} & & & \Lambda ^{2}_{2} & \\ \{ 0\} \ar [uur] \ar [rr] & & \{ 2\} & \{ 0\} \ar [uur] \ar@ {-->}[rr] & & \{ 2\} & \{ 0\} \ar@ {-->}[uur] \ar [rr] & & \{ 2\} . } \]

Here the dotted arrows indicate edges of $\Delta ^2$ which are not contained in the corresponding horn.

Example 1.1.2.12. The horns $\Lambda ^{1}_{0}$ and $\Lambda ^{1}_{1}$ are both isomorphic to $\Delta ^0$, and the inclusion maps $\Lambda ^{1}_{0} \hookrightarrow \operatorname{\partial \Delta }^1 \hookleftarrow \Lambda ^{1}_{1}$ induce an isomorphism $\Delta ^{0} \amalg \Delta ^{0} \simeq \operatorname{\partial \Delta }^1$.

Example 1.1.2.13. The horn $\Lambda ^{0}_{0}$ coincides with the $0$-simplex $\Delta ^{0}$.

Exercise 1.1.2.14. Let $0 \leq i \leq n$ be integers. For $j \in [n] \setminus \{ i\} $, we can regard the map $\delta ^{j}$ of Exercise 1.1.2.8 as a map of simplicial sets from $\Delta ^{n-1}$ to the horn $\Lambda ^{n}_{i} \subseteq \Delta ^{n}$. Show that, for any simplicial set $S_{\bullet }$, the construction

\[ ( f: \operatorname{\partial \Delta }^{n} \rightarrow S_{\bullet } ) \mapsto \{ f \circ \delta ^{j} \} _{j \in [n] \setminus \{ i\} } \]

determines an injection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{i}, S_{\bullet } ) \rightarrow \prod _{ j \in [n] \setminus \{ i\} } S_{n-1}$, whose image consists of ‘incomplete” sequences $( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _ n)$ satisfying $d_ j(\sigma _ k) = d_{k-1}( \sigma _{j})$ for $j, k \in [n] \setminus \{ i\} $ with $j < k$.