Here, again, is the complex we were looking at at the end of the last section:
Notice that there are two trees here which are, in a sense, unconstrained: the edge tree of the left-hand diagram, and the box tree of the right. On the left, this is simply because there is no diagram which "comes before" whose boxes would fix the structure of the edge tree. Similarly, on the right, there is no diagram which "comes after" whose edges would fix the structure of the boxes. We could have equally well drawn a different collection of edges on the left, and a different collection of boxes on the right without affecting the bonding relationship between the adjacent pairs.
In order to single out the opetopes among all possible complexes, we are going to fix these two ambiguities by specifying an initial and final condition. Hence we will say that an unlabeled opetopic complex is an opetope if
For example, the following complex is an opetope:
In fact, by demanding that the initial box tree be linear, we remove the need for the edges entirely, and hence, as shown above, we will simply leave them off since they do not correspond to any previous boxes.
In an opetope, each atomic diagram can be assigned a dimension, starting on the left and beginning with zero. Consequently, the above opetope is 5-dimensional. As we will see below, each box in the diagram itself corresponds to an opetope, and so we will often refer to the boxes generically as cells. When a particular opetope is under discussion, we will occasionally refer to the cells which comprise it as its faces.
Notice that, in view of our definition, each face appears exactly twice in the diagram: once in its proper dimension, and once in the following dimension as the edge which it corresponds to under the bond. This makes good geometric sense. Indeed, we expect an (n+1)-dimensional object to have a boundary which is n-dimensional and as we will see, the boundary of any one of our cells is easy to describe: for an (n+1)-dimensional cell, its boundary consists of exactly the n-cells corresponding to the edges which intersect the boundary of the box which represents it.
This observation provides us with a nice interpretation of what our atomic diagrams in fact depict: the boxes of a (positive dimensional) atomic diagram depict a collection of cells, while the edges are a picture of how these cells are glued together along cells of codimension 1.
Just as any face of a higher dimensional simplex is itself a simplex, and that of a higher dimensional cube a cube, every face of an opetope is also an opetope. The following interactive demonstration will serve to illustrate this point:
First of all, as you pass your mouse cursor over one of the cells, you will notice that a number of lower dimensional cells are highlighted. These are exactly the faces of the face you are pointing at. Futhermore if you click on one of the faces, its opetopic structure will be "extracted" into the bottom region, where you can verify that it also is an opetope in the sense defined above.
Constructing Opetopes Inductively
Our description of the opetopes provides a straightforward recipe for constructing them. Start with a finite, linear nesting of boxes. Draw the edge tree which corresponds to that nesting. Add boxes to taste, ensuring that there is a largest box enclosing all the others. Repeat.