In "Higher Topos Theory" by Lurie (Section 1.2.2, "Mapping spaces in Higher Category Theory") the notion of mapping space is defined as such: Given a simplicial set $S$ and two vertices $x,y\in S_0$, define $$\text{Hom}^R_S(x,y)$$ as the simplicial set with $n$-component equal to the simplicial maps $z:\Delta^{n+1}\rightarrow S$ such that $z|_{\Delta^{\{n+1\}}}=y$ and $z|_{\Delta^{\{0,\cdots,n\}}}$ is a constant simplex at the vertex $x$
I'm not sure what "constant simplex" at vertex means. I assume that $\Delta^{\{0,\cdots,n\}}$ means the sub-simplex of $\Delta^{n+1}$ generated by the the vertices $0,\cdots,n$ and "constant simplex" means that $z|_{\Delta^{\{0,\cdots,n\}}}$ factors through the map $x:\Delta^0\rightarrow S$ corresponding to $x$. Is this correct?
Yes, that's the correct interpretation. In other words, $\Delta^{\{0,\ldots,n\}}$ is maximally degenerate.