Identities and Units

Let us start with an object x in an opetopic category.

Then there is a canonical pasting diagram, the empty pasting diagram on x, which is represented by the trivial tree in dimension 1. This pasting diagram is depicted as follows:

According to the first axiom of an opetopic category, this diagram admits a target universal filler which looks as follows:

Moreover, applying the second axiom of an opetopic category, we learn that in fact, the composite, which we have called id-x, is in fact itself target universal.

In order to justify that the cell obtained in this manner behaves as an identity, let us suppose that we have an arrow f with source x:

We may then form the composite of f and id-x as follows:

If we now compose α with the filler obtained in the definition of id-x, we obtain the following:

Where the left universality of the resulting cell follows from the closure of target universal cells under composition. Taking a look at the resulting cell, we see that we have constructed a left universal cell connecting f and f ∘ id-x, whose shape is simply that of a 2-glob:

We interpret this cell as the right unit law of our higher category. In the next section, we will justify the intuition that this cell is in fact an equivalence between f and f ∘ id-x.