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Cloudifold (May 01 2022 at 15:25):I cannot understand (both intuitively and formally) why Steenrod Algebra is generated by Steenrod Squares.
Tim Campion (May 03 2022 at 15:04):Some would define "the Steenrod algebra" to be generated by Steenrod squares. How are you defining the Steenrod algebra?
Ulrik Buchholtz (May 03 2022 at 16:59):I'm guessing the default definition would be the (homotopy groups of the) endomorphism ring spectrum of ?
Tim Campion (May 03 2022 at 17:07):That does seem to be a reasonable guess!
Personally I have very little intuition for how the Steenrod algebra works, but the dual Steenrod algebra somehow makes a lot more sense to me. It has a natural interepretation in terms of formal group laws, and its comultiplication is somehow much easier to think about than the multiplication of the Steenrod algebra (even though they encode exactly the same information!). So the best I could say about "why the Steenrod squares generate the Steenrod algebra" is "you can read off the dimensions of the graded pieces of the Steenrod algebra from the much more natural and easily-explained dual Steenrod algebra, and it so happens that the Steenrod squares give you "enough" elements to generate it".
The closest I've personally come to understanding how the Steenrod squares themselves work was from some notes by Lurie. It still seems like magic to me that he only needs a very small part of the Eoo structure on HF2 to construct all the Steenrod squares... I once tried to understand the universal property discussed in Ch VIII.2 of the Hoo book but I didn't quite succeed.
Cloudifold (May 10 2022 at 12:06):Merci, I would take a read.
Cloudifold (Aug 02 2022 at 05:48):I also found a modern approach to Steenrod operations : In Chapter 4, IV 1 of is article On topological cyclic homology.