TITLE: Generalized graph theory
QUESTION [13 upvotes]: This question may be kind of 'out there' but it got me thinking.
In graph theory we have a set of vertices $V$ and a set of edges $E$ which is made up of 2-element subsets of $V$ (either unordered or ordered).
Is there any notion in mathematics that considers a sort-of generalized graph, consisting of vertices $V$ and a set of 'edges' which is made up of, say, 3-element subsets of $V$ -- planes, instead of lines in the 2-element case. What about 'graphs' with 'edges' that are $n$-element subsets of $V$?
Does this notion even have any value?

REPLY [6 votes]: If you're looking for a generalization of graphs that has a more geometric flavour, CW complexes come to mind.
http://en.wikipedia.org/wiki/CW_complex
A graph is a 1 dimensional CW complex.  You have a discrete set of points, a set of line segments, and maps that attach the boundary of line segments to points (i.e. source and target).
However, CW complexes generalize to n dimensions.  Here you could have a 2 dimensional complex that attaches the boundary of a higher order "edge" to "normal edges."  Then the boundary of the boundaries would give you a set of "vertices."
CW complexes play a large role in homotopy theory.