most recent 30 from http://mathoverflow.net 2010-03-16T05:31:17Z http://mathoverflow.net/feeds/question/6346 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6346/simplicial-set-notation-and-vocabulary-question fpqc 2009-11-21T03:13:50Z 2009-11-21T19:54:55Z

<p>Notation question:</p> <p>What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? <strong>UPDATE</strong>: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.</p> <p>Vocabulary question:</p> <p>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 $x$. </p> <p>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. </p> <p>I've searched through a number of books on homotopy theory, algebraic topology, etc. and I've been unable to find these precise usages. </p> <p>I ask these questions only because I'm reading HTT by Lurie, and these usages come up and they're quite confusing.</p>

http://mathoverflow.net/questions/6346/simplicial-set-notation-and-vocabulary-question/6349#6349 Scott Carnahan 2009-11-21T04:14:22Z 2009-11-21T19:54:55Z

<p>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.</p> <p>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.</p> <p><b>Edit in response to comment:</b> 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 X_i 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 X_n.</p>