Progress towards semi-strictification for tetracategories

Adrian Miranda – 31 August 2022

In previous talks I outlined how strictification in dimensions 2 and 3 can be broken into several steps. 1. Read off generating data (ie a directed graph/2-computad) from the weak-n-category (ie bicategory/tricategory) A. 2. Freely construct the fragment of the semi-strict n category containing only k < n dimensional cells (ie category/sesquicategory). 3. Form a discrete semi-strict n category on that, call it D(A). 4. Pick an arbitrary but consistent way of evaluating n-1 cells. 5. Use the evaluation to form a weak n-functor Ev: D(A) --> A. 6. Factorise Ev = F^ B with F fully faithful on n-cells and B bijective on k < n -cells. This week I will share the progress I have made in executing this program in the case n = 4. Briefly, steps 1, 2, 3, and 4 are done, while steps 5 and 6 need some more work. That being said, I'll also be able to explain some strategies that I've been trying to use to make those steps work.