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Oct 31, 2022 at 13:29 comment added daniel gratzer To follow up on this @TimCampion's earlier comment, Cisinski and Nguyen have released their fully quasi-categorical proof of straightening-unstraightening: arxiv.org/abs/2210.08945
Dec 20, 2021 at 15:23 comment added Vanni There is also this new proof of the straightening equivalence by Hebestreit, Heuts and Ruit which is much shorter than the one in HTT.
Feb 10, 2021 at 18:46 comment added Tim Campion I'll also point to a great simplification (at least for many purposes) which Rune Haugseng and Dylan Wilson pointed out to me of the technique developed in 4.2.3.10 of HTT for decomposing (co)limit diagrams.
Feb 10, 2021 at 18:42 comment added Tim Campion I hope people will keep adding things, but one thing to mention is that Cisinki and Nguyen (perhaps still in progress? see his thesis and this) have made progress on simplifying the straightening-unstraightening construction, with the goal of performing it "internally" to the quasicategory model rather than passing back and forth with simplicial categories.
Sep 24, 2018 at 4:24 history edited Tim Campion CC BY-SA 4.0
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Sep 24, 2018 at 1:22 history answered Tim Campion CC BY-SA 4.0