Notation question:

What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? UPDATE: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.

Vocabulary question:

Suppose $z:\Delta^{n+1} \rightarrow S$ is a morphism of simplicial sets. What does the following translate to in algebraic terms: $z|\Delta^{ \{0,\ldots,n \} }$ is a constant simplex at a vertex $x$.

So mainly, I just don't know what that is supposed to mean, "is a constant simplex at the vertex x". Everything else makes fine sense.

I've searched through a number of books on homotopy theory, algebraic topology, etc. and I've been unable to find these precise usages.

I ask these questions only because I'm reading HTT by Lurie, and these usages come up and they're quite confusing.