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

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

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

]]>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.

]]>apparently they brought their point of view into final form in

- Keith Hardie, Howard Marcum, Nobuyuki Oda,
*Bracket operations in the homotopy theory of a 2-category*, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001)

Unfortunately, that’s the one article in the series that I don’t find any electronic trace of…

]]>Also equation (2.2) in

- Keith Hardie, Klaus Heiner Kamps, Howard Marcum,
*The Toda bracket in the homotopy category of a track bicategory*, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

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

]]>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)

]]>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.

]]>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.

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