Have been thinking a lot lately about the role of simplices in n-category theory. I'm well aware that some definitions of weak n-category depend on them heavily. That seems strange to me, for while the n-simplex clearly is an n-dimensional thingy, and while people have methods of giving "direction" to all the cells, it seems to only be naturally directed along the 1-cells.
That is, if I ask for a map from an n-simplex to an m-simplex, I only need to talk about conditions on how you map the vertices to one another, (i.e. an order-preserving function from the totally ordered set (0...n) to the totally ordered set (0...m)) and there are no more choices to be made.
In other words the n-simplex is telling me how I may compose n arrows in a row. The 0-simplex, a point, gives identity maps. The 1-simplex is an arrow. The 2-simplex, a triangle, tells me how to compose 2 arrows and get a third (and the 2-cell it contains ought to be invertible to express that the path consisting of the two composed arrows is equal to its composite) and the 3-simplex, a tetrahedron, should tell me that composition is associative.
None of these geometric/combinatorial objects actually seems to say anything about general composition of directed higher cells, only perhaps composition of undirected ones as needed for coherence. I feel the need for another infinite family of "simplicial sequences" that may not in fact be linear sequences, since there is a highly branching collection of ways of composing many 2-cells.