# Contents

## Idea

In as far as the simplicial $n$-simplex ${\Delta }^{n}$ (a simplicial set) is a combinatorial model for the $n$-ball, its boundary $\partial {\Delta }^{n}$ is a combinatorial model for the $\left(n-1\right)$-sphere.

## Definition

The boundary $\partial {\Delta }^{n}$ of the simplicial $n$-simplex ${\Delta }^{n}$ is the simplicial set generated from the simplicial set ${\Delta }^{n}$ minus its unique non-degenerate cell in dimension $n$.

This may equivalently be described to be degreewise the coequalizer

$\coprod _{0\le i\coprod_{0\le i\lt j\le n}\Delta[n-2]\rightrightarrows\coprod_{0\le i\le n}\Delta[n-1]\to \partial \Delta[n]

defined by the (induced coproduct maps of the) simplicial identities ${d}_{i}{d}_{j}={d}_{j-1}{d}_{i}$.

Regarding ${\Delta }^{n}$ as the presheaf on the simplex category that is represented by $\left[n\right]\in \mathrm{Obj}\left(\Delta \right)$, then this means that $\partial {\Delta }^{n}$ is the simplicial set generated from $\Delta$ minus the identity morphism ${\mathrm{Id}}_{\left[n\right]}$.

There is a canonical monomorphism

${i}_{n}:\partial {\Delta }^{n}↪{\Delta }^{n}\phantom{\rule{thinmathspace}{0ex}},$i_n : \partial \Delta^n \hookrightarrow \Delta^n \,,

the boundary inclusion .

The geometric realization of this is the inclusion of the $\left(n-1\right)$-sphere as the boundary of the $n$-disk.

Simplicial boundary inclusions are one part of the cofibrant generation of the classical model structure on simplicial sets.

## Examples

For low $n$ the boundaries of $n$-simplices look as follows (see also the illustrations at oriental)

• $\partial {\Delta }^{0}=\varnothing$;

• $\partial {\Delta }^{1}=\partial \left\{0\to 1\right\}=\left\{0,1\right\}={\Delta }^{0}\bigsqcup {\Delta }^{0}$;

• $\partial {\Delta }^{2}=\partial \left\{\begin{array}{ccc}& & 1\\ & ↗& ⇓& ↘\\ 0& & \to & & 2\end{array}\right\}=\left\{\begin{array}{ccc}& & 1\\ & ↗& & ↘\\ 0& & \to & & 2\end{array}\right\}$

Revised on March 12, 2012 21:14:27 by Urs Schreiber (89.204.153.191)