There is also a notion of polyhedral -complex, defined in (Jones, 1983). There are no degeneracies in this theory, but it does allow for the shapes quite, but not completely, general forms of regular cell decompositions of cells. This gives a solution to the problem of defining general compositions. One has to define:
What are the pieces that might be composable?
When are they composable?
What is their composite?
What are the axioms on the composition?
On the face of it, the last problem seems the hardest. It turns out that the last two -complex axioms are sufficient! Thus the geometry determines the algebra.