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I have re-written the Idea-section of this entry, trying to bring out clearly the simple idea that the Toda bracket is just the pasting composite of two overlapping null homotopies.
Also I added pointer to the original reference by Toda (though still need to dig out page and verse where his bracket is actually defined).
I was gonna do more to this entry, but now I am out of steam. Maybe later.
I have now expanded the re-written Idea-section a little further (here).
In particular I have added a sequence of homotopy-commutative pasting diagrams which shows how the simple abstract-homotopy/$\infty$-category theoretic construction reduces to the popular 1950s-style description via consecutive “extensions over cones”.
I have been scanning the literature to find an author who would present not the 1950s-style but this 2010s-style version of the definition of the Toda bracket; but so far I haven’t found any such reference. If somebody has a pointer, please let me know.
found something:
In
the homotopy-pasting-diagram formulation of the Toda bracket is equations (0.2)-(0.3).
(insisting on some double-categorical structure, but still)
Also equation (2.2) in
I see now that Hardie & Kamps et al. have a series of articles on this aspect. Will add them…
apparently they brought their point of view into final form in
Unfortunately, that’s the one article in the series that I don’t find any electronic trace of…
Re #5: The article is available here: https://rendiconti.dmi.units.it/volumi/33/02.pdf
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