nLab
boundary of a simplex

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 $(n-1)$-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

$$\coprod _{0\le i<j\le n}\Delta [n-2]\rightrightarrows \coprod _{0\le i\le n}\Delta [n-1]\to \partial \Delta [n]$$\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 $[n]\in \mathrm{Obj}(\Delta )$, then this means that $\partial {\Delta }^n$ is the simplicial set generated from $\Delta$ minus the identity morphism ${\mathrm{Id}}_{[n]}$.

There is a canonical monomorphism

$$i_n:\partial {\Delta }^n\hookrightarrow {\Delta }^n,$$i_n : \partial \Delta^n \hookrightarrow \Delta^n \,,

the boundary inclusion .

The geometric realization of this is the inclusion of the $(n-1)$-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)

• $\partial {\Delta }^0=\varnothing$;

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

• $\partial {\Delta }^2=\partial \left\{\begin{array}{ccc}& & 1\\ & \nearrow & \Downarrow & \searrow \\ 0& & \to & & 2\end{array}\right\}=\left\{\begin{array}{ccc}& & 1\\ & \nearrow & & \searrow \\ 0& & \to & & 2\end{array}\right\}$

Related concepts

• horn, spine