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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.

• 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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 11th 2021

found something:

In

• 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)

• 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…

• 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…

• 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.

• 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.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 11th 2021

Re #5: The article is available here: https://rendiconti.dmi.units.it/volumi/33/02.pdf

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeFeb 11th 2021

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