The standard $n$-simplex is defined as the simplicial set $\Delta^n$ with $\Delta^n_m = \text{Hom}_\Delta([m],[n])$. The boundary of $\Delta^n$ can be defined as the simplicial set $\partial \Delta^n$ with $\partial \Delta^n_m$ the set of all maps from $[m]$ to $[n]$ that are not surjective.
In searching through the literature I was not able to find a reference that defines the boundary of a simplicial set in general. So I was wondering if some of you have a definition (or a reference)? And why is your definition a (or the) ''correct'' one?
I was thinking of the following. Let $X$ be a simplicial set, then define the boundary of $X$ to be the simplicial set $\partial X$ generated by all simplices $x\in X_n$ such that there is a unique non-degenerate $y\in X_{n+1}$ with $d_i(y) = x$ for some $i$. Let the face and degeneracy maps be inherited from $X$. What do you think of this definition?