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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2021

    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.

    diff, v12, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    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.

    diff, v14, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    found something:


    • K. A. Hardie, K. H. Kamps, and R. W. Kieboom, Higher homotopy groupoids and Toda brackets, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 (euclid:hha/1139840198)

    the homotopy-pasting-diagram formulation of the Toda bracket is equations (0.2)-(0.3).

    (insisting on some double-categorical structure, but still)

    diff, v15, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    Also equation (2.2) in

    I see now that Hardie & Kamps et al. have a series of articles on this aspect. Will add them…

    diff, v15, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021
    • (edited Feb 11th 2021)

    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…

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    Just trivia, but I have made explicit (here) how the Toda bracket is not a function with “indeterminate” values, as traditional nomenclature still insists, but a bundle with fibers.

    diff, v17, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    I have made explicit (here) the group action on the set of choices of Toda brackets, which becomes manifest from the homotopy-pasting perspective.

    diff, v18, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 11th 2021

    Re #5: The article is available here:

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2021

    Thanks! That’s great. I have added the link to the entry.

    diff, v19, current