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Christopher Tapo (Sep 12 2023 at 22:32):One of my projects has been following this line of questioning:
"If the associahedra underlie associativity axioms for (weak) n-categories, and the multiplihedra/functoriahedra underlie axioms for (weak/lax) n-functors, what combinatorial structures underlie the axioms for higher morphisms/transfors like transformations or modifications? Are they polytopes or polytopal complexes? Could knowing this combinatorial structure be used for operadic/computadic methods for coherence?"
Currently, I'm working on extending and generalizing the use of polygraphs/rewriting for coherence for bicategories and their transfors appearing in the work of Maxime Lucas (https://arxiv.org/pdf/1508.07807.pdf) to coherence for tricategories and their transfors. The work of Simon Forest (https://arxiv.org/pdf/2109.05369.pdf) would also need to be involved to cover the semistrict/Gray parts of coherence for tricategories. The goal is to work on coherence for higher categories using movies of 2D string diagrams rather than surface diagrams or higher dimensional versions of string diagrams.
Here I've included an axiom for tritransformations using polygraphs that can be compared to the version appearing in Nick Gurski's book "Coherence in Three-Dimensional Category Theory":
Would be interested to hear thoughts on this!