Geometry

In this section, we explore the connection between our opetopic diagrams and the more traditional geometry of pasting diagrams in low dimensions. In each example, hover over the cells of the opetopic diagram to highlight the part of the pasting diagram it corresponds to.

Dimension 0

As usual, dimension 0 is rather uninteresting: there is a single opetope of dimension 0 corresponding to a point.

The Object

Dimension 1

In dimension 1, there is also a unique opetope, this time corresponding to an arrow.

The Arrow

Dimension 2

Dimension 2 becomes more intersting: we already have infinitely many opetopes of dimension two, one for each natural number which counts the number of source arrows.

The Drop

Drops, like the one above and its higher dimensional counterparts play the role of degeneracies for opetopes.

The Two-Glob

This first globular shape is an example of a more general phenomenon: all the higher dimensional globs are also opetopes.

The Simplex

In contrast to the situation for globs, the two-dimensional simplex is the only overlap with the simplicial world: no higher dimensional simplex is also an opetope.

The Quadrilateral

We hope the reader can see how the pattern continues on from here.

Dimension 3

Things become much more interesting in dimension 3, as we hope the following example will start to show.

A Three Cell

As you can see, already in dimension 3, the naive geometric picture begins to fail us: the boundary arrows and objects must be duplicated in order to make room for the target cell. You are supposed to imagine that the pasting diagram on the right is slid on top of its target two cell, with the edges identified, and that the top cell Φ represents the space in between when the pasting diagram is bent out of the page.

The opetopic diagram, however, presents no such ambiguity. Each cell occurs exactly once in the dimension where it lives.