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 setgenerated 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

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 $Id_{[n]}$.