We will say that an opetopic set is an opetopic category if the following two conditions are satisfied:
Note that we continue the convention begun in the second about universal properties where the cells or properties which are the conclusions of these axioms are depicted in red.
Comparison with Baez-Dolan Opetopic Categories
We note that the second axiom here is strengthened with respect to that given by Baez and Dolan in their original definition. Their definition requires only that composites of target universals are also target universal, that is, they require our axiom only in the case where it is the cell w which is not already known to be target universal, as depicted by the following diagram:
Readers familiar with model categories will recognize this as an opetopic adaptation of the 2-for-3 axiom of equivalences in a model category. We think this makes the axiom reasonable from a theoretical point of view. In compensation for this strengthening of the axioms for a category, our definition of universal properties are weaker that those given by Baez-Dolan. We will show later that under this stronger definition of category, we can recover the full strength of Baez-Dolan style universal cells.