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David Michael Roberts (Jul 30 2020 at 08:52):Maybe we have finally tracked it down...
https://arxiv.org/abs/2007.14505
Finally, we prove a version of the homotopy hypothesis for the infinity-groupoids in the weak model, and exhibit a specific mistake in a proof by Kapranov and Voevodsky that had previously been refuted indirectly.
Though I seem to think there was another paper not too long ago that also claimed to have found the specific mistake.
Amar Hadzihasanovic (Jul 30 2020 at 09:25):Hi, a while ago we had a discussion about this here, in this thread :)
I've known what the error was for a while, but that discussion made me realise that there could be some interest in having it published somewhere. So I wrote it down in this paper.
Amar Hadzihasanovic (Jul 30 2020 at 09:26):Anyway, I'll take the chance to say that I'd be happy to get comments or feedback on the paper!
Amar Hadzihasanovic (Jul 30 2020 at 09:34):This started as a revision of an older draft but then there was a lot of new content, a lot of old content that didn't fit anymore, and I made it a new one.
The main purpose is to give a framework for “higher-dimensional rewriting”, by which I mean broadly all sorts of diagrammatic manipulations (such as “cutting and pasting” diagrams) in all dimensions, which is nearly as expressive as polygraphs/strict higher categories but avoids the pitfalls of strictness (unsoundness for homotopical algebra).
So I hope it can be useful to applied CT when it is forced to go higher than dimensions 2 or 3, with their safety net of coherence and strictification theorems.
Amar Hadzihasanovic (Jul 30 2020 at 09:41):It's a long paper, so when I have time I'd like to write a brief “user's manual” on how to formalise diagrams in this framework. Maybe in time for the next ACT conference...