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

1.2.4 Horns

We now consider some elementary examples of simplicial sets which will play an important role throughout this book.

Construction (The Horn $\Lambda ^{n}_{i}$). Suppose we are given a pair of integers $0 \leq i \leq n$ with $n > 0$. 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 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 Remark

Example The horn $\Lambda ^{1}_{0} \subset \Delta ^1$ is the vertex $\{ 1\} $, and the horn $\Lambda ^{1}_{1} \subset \Delta ^1$ is the vertex $\{ 0\} $ (see Example In particular, $\Lambda ^{1}_{0}$ and $\Lambda ^{1}_{1}$ are abstractly isomorphic to the standard $0$-simplex $\Delta ^{0}$. Moreover, the boundary $\operatorname{\partial \Delta }^{1}$ is the disjoint union of $\Lambda ^{1}_{0}$ and $\Lambda ^{1}_{1}$.

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

\[ \xymatrix@R =20pt@C=20pt{ & \{ 1\} \ar@ {-->}[ddr] & & & \{ 1\} \ar [ddr] & & & \{ 1\} \ar [ddr] & \\ & \Lambda ^{2}_{0} & & & \Lambda ^{2}_{1} & & & \Lambda ^{2}_{2} & \\ \{ 0\} \ar [uur] \ar [rr] & & \{ 2\} & \{ 0\} \ar [uur] \ar@ {-->}[rr] & & \{ 2\} & \{ 0\} \ar@ {-->}[uur] \ar [rr] & & \{ 2\} . } \]

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

Remark Let $0 \leq i \leq n$ be integers with $n > 0$. Then the horn $\Lambda ^{n}_{i}$ is connected. If $n = 1$ or $n=2$, this follows by inspection (see Examples and For $n \geq 3$, the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is bijective on simplices of dimension $\leq 1$, so the desired result follows from Proposition (together with the connectedness of the standard simplex $\Delta ^{n}$; see Example

Remark Let $0 \leq i \leq n$ be integers with $n > 0$. It follows from Proposition that the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ induces a homeomorphism from the geometric realization $| \Lambda ^{n}_{i} |$ to the closed subset of $| \Delta ^{n} |$ given by

\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j \neq i$} \} . \]

Let $n$ be a positive integer. For every pair of distinct integers $i,j \in [n]$, the inclusion map $\delta ^{j}_{n}$ of Construction can be regarded as a morphism of simplicial sets from $\Delta ^{n-1}$ to the horn $\Lambda ^{n}_{i}$. We have the following counterpart of Proposition

Proposition Let $0 \leq i \leq n$ be integers with $n > 0$. For any simplicial set $S_{\bullet }$, the map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{i}, S_{\bullet } ) \rightarrow (S_{n-1})^{n} \quad \quad f \mapsto \{ f \circ \delta ^{j}_{n} \} _{0 \leq j \leq n, j \neq i} \]

is an injection, whose image is the collection of “incomplete” sequences

\[ ( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _ n) \]

which satisfy the identity $d^{n-1}_ j(\sigma _ k) = d^{n-1}_{k-1}( \sigma _{j})$ for $j, k \in [n] \setminus \{ i\} $ with $j < k$.

Proof. We proceed as in the proof of Proposition, with minor modifications. Set $Q = [n] \setminus \{ i\} $ and let $w: \coprod _{\ell \in Q} \Delta ^{n-1} \rightarrow \Lambda ^{n}_{i}$ be the map given on the $\ell $th summand by $\delta ^{\ell }_{n}$. To prove the first assertion of Proposition, 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 $\Lambda ^{n}_{i}$ if and only its image does not contain $Q$: that is, if and only if there exists some integer $j \in Q$ such that $\alpha $ factors through $[n] \setminus \{ j \} $. In this case, there is a unique $m$-simplex $\beta _{j}$ which belongs to the $j$th summand of $\coprod _{\ell \in Q} \Delta ^{n-1}$ and satisfies $w( \beta _ j ) = \alpha $.

For every integer $\ell \in Q$, let $u_{\ell }: \coprod _{j \in Q, j < \ell } \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ be the map given on the $j$th summand by $\delta ^{j}_{n-1}$, and let $v_{\ell }: \coprod _{k \in Q, k > \ell } \Delta ^{n-2} \rightarrow \Delta ^{n-1}$ be the map given on the $k$th summand by $\delta ^{k-1}_{n-1}$. Passing to the coproduct over $\ell $ and reindexing, we obtain a pair of maps

\[ (u,v): \coprod _{j,k \in Q, j < k} \Delta ^{n-2} \rightrightarrows \coprod _{\ell \in Q} \Delta ^{n-1}. \]

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, and therefore factors uniquely through a map $\overline{w}: \operatorname{Coeq}(u,v) \rightarrow \Lambda ^{n}_{i}$. Proposition 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 $\Lambda ^{n}_{i}$. 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 $j,k \in Q$ which do not belong to the image of $\alpha $, then $\beta _{j}$ and $\beta _{k}$ have the same image in $\operatorname{Coeq}(u,v)_{\bullet }$. If $j = k$, this is automatic; we may therefore assume without loss of generality that $j < k$. In this case, the desired result follows from the observation that we can write $\beta _ k = u(\gamma )$ and $\beta _ j = v(\gamma )$, where $\gamma $ is the $m$-simplex of the $(j,k)$th summand of $\coprod _{j,k \in Q, j < k} \Delta ^{n-2}$ corresponding to the nondecreasing function $[m] \xrightarrow {\alpha } [n] \setminus \{ j < k \} \simeq [n-2]$. $\square$