Opetopic Complexes

In their original paper introducing the opetopes, Baez and Dolan described a notation which they called metatree notation for working with them. This notation was then refined and improved in an article of Joyal, Kock, Batanin and Mascari. The notation developed here is essentially that of the latter paper, with some minor modifications which we will point out along the way.

As mentioned in the introduction, opetopes can be seen as certain well-formed higher dimensional trees. It may not be surprising, therefore, that we begin by fixing some notation for trees. Opetopic diagrams will rely on the combination of two methods of depicting a rooted tree. An example of each of these methods is shown in the following diagram.

As the reader can easily verify, both diagrams represent the same rooted tree. Of the two, the picture on the right is likely the more familiar: the tree is depicted as a (partial) graph consisting of nodes and edges. The diagram on the left, by contrast, employs a collection of boxes, with their containment representing the tree structure. The correspondence between the two can be summarized by observing that our labelling specifies a bijection between the boxes on the left, and the edges on the right.

There is a certain geometric intuition which accompanies these two representations of our tree. Specifically, each can be regarded as a kind of projection, with the collection of boxes representing our view "from above" and the edges "from the side". From a slightly different perspective, as in the diagram below, the bijection between our boxes and edges is rendered as their intersection in a higher dimensional space.

It is worth remarking that the view of our tree as a collection of boxes makes no mention of the tree's "nodes", which are central to its representation as a graph. The diagram above makes clear that we can regard these nodes as a representation of the space, which is implicit, in between different levels of boxes. We will return to this theme later in our discussion.

Atomic Diagrams

Next, consider the following diagram which, in contrast to the situation above, is not a representation of a single tree in two different ways, but rather a representation of two different trees which share some structure.

Observe that the boxes shaded grey play a dual role: they are simultaneously acting as the "leaves" of the tree formed by the boxes (since they contain no boxes themselves), while at the same time serving as the "nodes" for the tree determined by the edges. (Here our presenation differs slightly from that of KJBM. The signifigance of this modification will become more clear in our section about geometry).

We refer to diagrams such as the one above as atomic diagrams.

Returning to the perspective view from above, we see that one way to understand an atomic diagram is as two trees drawn at right angles to one another in the sense that one tree is depicted "vertically" using the axis perpendicular to your screen, while the other is depicted "horizontally" lying in the plane of your screen.

When we need to distinguish between these two trees, will will refer to them as the box tree and the edge tree.

Atomic diagrams are subject to two rules in order to be considered well formed:

The interior of every box is non-empty connected subtree of the edge tree.
The box and edge trees are each rooted

The first condition prevents the following two situations, which from now on we regard as pathological:

On the left, the box in red encloses a subtree of the edge tree which is disconnected, whereas on the right, the red box is "floating" and not connected to any part of the edge tree.

The meaning of the second condition applied to the edge tree should be clear enough. For the box tree, it simply asserts that there is a largest box containing all the others. We will see later on that this condition may be relaxed in a certain sense, but for now, all trees under consideration will be rooted.

Bonds

As we have seen, each atomic diagram is a representation of two trees simultaneously. It stands to reason, therefore, that the following diagram represents four trees.

Of course, this is in fact the case. But it is not the whole story. Indeed, by toggling the switch above (which removes the edge tree from the left diagram and the box tree from the right) we see that the two remaining trees are in fact isomorphic.

We will say that two adjacent atomic diagrams with such a specified isomorphism are bonded.

Complexes

A opetopic complex is simply a sequence of atomic diagrams in which adjacent diagrams are bonded in the sense of the last section. For example, with the aid of the buttons below, the reader can verify that each of the two adjacent pairs in the following sequence is bonded.

Briefly, then, an opetopic complex is a sequence of atomic diagrams in which the boxes of one diagram are in bijection with the edges of the next.