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Tim Hosgood (May 09 2021 at 15:30):I've posted a question on the nForum, because it came from me wanting to be able to edit the page, but I guess it's a good idea to maybe ask it here too.
Three things which have similar names are the following: twisting cochains, twisting cochains, and twisted complexes (you might notice that the first two have, in fact, identical names...). The first is the thing defined at https://ncatlab.org/nlab/show/twisting%20cochain ; the second is the thing defined by Toledo–Tong (I'll attach an image because I'm too lazy at the moment to type out the definition) (but the definition can also be found in eg the slides from a talk I gave, linked further down); and the third is the thing defined by Bondal–Kapranov (https://ncatlab.org/nlab/show/twisted+complex).
Screenshot-2021-05-09-at-16.27.19.png
Somehow these three should all be either equivalent or specific examples of one another. Damien Calaque bestowed upon me a beautiful explanation of how twisting cochains (in the sense of Toledo–Tong) are (strictly) specific examples of twisted complexes, and so this is written down in my thesis, but I don't know how common it is as folklore. I'm interested in this other definition, which I've never really studied before. Is it a generalisation of the Bondal–Kapranov one, or is it a strictly different generalisation of the Toledo–Tong one?
Tim Hosgood (May 09 2021 at 15:31):(I gave a talk at YTM19 where I explained a bit about the Toledo–Tong story and the Bondal–Kapranov story, and showed how the latter generalises the former, so maybe the slides might be helpful for some people: https://github.com/thosgood/talks/blob/master/twisting-cochains-YTM19/twisting-cochains-YTM19.pdf)
Tim Hosgood (May 09 2021 at 15:32):(in this talk I actually talk a bit about the nLab definition, but don't say anything at all about how it relates to the other two, because I didn't know!)
Tim Hosgood (May 09 2021 at 15:34):one reason that it would be really nice to understand this piece of the puzzle is that each definition has a seemingly very different motivation! I know the main motivations for Toledo–Tong and Bondal–Kapranov, and, from what I gather, the nLab definition is to do with factoring through the Yoneda embedding
John Baez (May 09 2021 at 15:56):Twisting, twisting... twisting the night away