Doubly degenerate Trimble 3-categories and braided monoidal categories

Eugenia Cheng – 16 August 2017

Trimble's approach to defining weak $n$-categories is by iterated enrichment, where each stage of enrichment is parametrised by the action of an operad. For some time it was thought that this definition would be ``too strict'' because although associativity and units are made weak, interchange remains strict. However, work by Joyal and Kock has shown that an alternative strictification to the usual one is possible: interchange can be made strict if units and associativity are not. Specifically, they proved that, given any monoidal 2-category with this level of weakness, End(I) is a braided monoidal category, and that every braided monoidal category arises in this way, up to some suitable equivalence.

We will give an analogous comparison for doubly degenerate Trimble 3-categories. We will construct a comparison functor to braided monoidal categories inspired by Joyal and Kock's use of cliques, and show that every braided monoidal category arises in this way, up to a suitable notion of equivalence. There is a difficulty in that notion of equivalence, as Trimble's original definition cleverly sidesteps the issue of weak functors, whereas we will need it. Thus we also propose a definition of weak functor for doubly degenerate Trimble 3-categories, based on the work of Cheng and Gurksi showing that an approach using icons gives the right higher maps between degenerate n-categories. Using these weak functors we construct a 2-category of doubly degenerate Trimble 3-categories and give a sense in which this is 2-equivalent to a 2-category of braided monoidal categories.