Extrusions

We will be interested in this section in two constructions which can be made on an arbitrary cell, embedding it in a diagram of the next higher dimension.

We first fix some notation. Consider the diagram below:

Recall that according to our definition, an opetope must finish with a single cell in the top dimension. In view of the bonding relationship, this implies that the codimension 1 part of the diagram consists of a single box, the target of the cell, as well as an arbitrary tree consisting of its sources.

We will refer to the top dimensional cell, together with its codimension 1 part as the head of the diagram as indicated. Moreover, to depict a cell of arbitrary dimension, we will depict just the head, using an ellipsis to denote the lower dimensional part.

With the above considerations established, the following is meant to indicate a cell f of some arbitrary dimension with its codimension 1 faces depicted as well.

Notice that we must choose some tree in our diagram to serve as the source tree for the cell f, but it will be clear from what follows that our constructions do not depend in any way on the shape of this tree.

Target Extrusion

Our first construction is called a target extrusion and is demonstrated in the interactive display below.

Notice that the extrusion consists of two steps: first, enclose the target in a new box, modifying the edge tree of the next dimension to accomodate the bonding relation. Next, enclose the remaining tree and add a new top dimensional cell.

Source Extrusions

Source extrusions are similar, but work with respect to a given source cell. In the following demonstration, we perform a "source extrusion at x". It should be clear that the same construction can be applied equally well to any other chosen source face.

Observe that in the source extrusion, we add a new cell "above" the chosen source face, taking care to maintain the bonding relationship, and then enclosing the resulting diagram with a target box, resulting in a new opetopic diagram.