Simplicial set notation and vocabulary question.

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.

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Check for yourselves on Google, there are a total of 4 unique results for "simplex at the vertex" and 7 unique results for "simplex at a vertex". – Harry Gindi Nov 21 '09 at 3:49
Where do you see it in HTT? – S. Carnahan Nov 21 '09 at 3:58
I've removed the offending remark. Anyway, Scott Carnahan, see page 24 for the question about notation and see page 27 for "constant simplex at the vertex x". Also, David Speyer has fixed the LaTeX issue, so I will remove the no-longer relevant remarks. – Harry Gindi Nov 21 '09 at 4:06

1 Answer

Okay, I've found the relevant notation in Higher Topos Theory. The first is at the end of 1.1.5.10, and is the simplicial set of maps from an n-1 element set to the 1-simplex (i.e., an n-1-cube). The second is at the end of warning 1.2.2.2, and describes a constant map of simplicial sets whose image is x. The curly braces give a reference to the specific ordered set that defines the the simplex.

A simplex at a vertex is a degenerate simplex. You take the simplicial subset defined by x, and demand that the map from your simplex to the target factor through the inclusion of x.

Edit in response to comment: You can think of vertices in (at least) two ways. One way is as an element of S0, i.e., a zero-simplex of the simplicial set. Another way is as a simplicial subset X of S, such that X0 is the chosen element of S0, and all Xi have a single element, namely the image of X0 under the unique degeneracy map. The statement is that the map Z takes a particular nondegenerate n-dimensional face of $\Delta^{n+1}$ to the unique element of Xn.

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For the n-1-cube part, are we assuming that the n-1 element set is ordered or no? – Harry Gindi Nov 21 '09 at 4:20
It is ordered, but it doesn't matter. We're looking at maps from an ordered set to a simplex, and there is no condition on orientation. – S. Carnahan Nov 21 '09 at 4:33
Two questions: What do you mean by "simplicial subset defined by x"? Whatever that means, let's let this "simplicial subset defined by x" be called X. Then did you mean that a "simplex at x" is a map that factors through the inclusion of x or the inclusion of X? – Harry Gindi Nov 21 '09 at 18:34
Much thanks for the help. – Harry Gindi Nov 21 '09 at 20:28