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

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

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 $[n] \in Obj(\Delta)$, then this means that $\partial \Delta^n$ is the simplicial set generated from $\Delta$ minus the identity morphism $Id_{[n]}$.

There is a canonical monomorphism

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

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)

• $\partial \Delta^0 = \emptyset$;

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

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

Related concepts

• horn, spine