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Expanmded the idea section at cup product
Need to add these two references (does the nLab have a separate article for Steenrod’s cup-i products?):
https://arxiv.org/abs/2010.02571
Cochain level May-Steenrod operations
Ralph M. Kaufmann, Anibal M. Medina-Mardones
Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-i products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, the need to have an effectively computable definition of Steenrod operations has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-i products on the simplicial and cubical cochains of spaces.
https://arxiv.org/abs/1810.06505
An axiomatic characterization of Steenrod’s cup-i products
Anibal M. Medina-Mardones
We show that any construction of cup-i products on the normalized chains of simplicial sets is isomorphic -- not just homotopic -- to Steenrod's original construction if it is natural, minimal, non-degenerate, irreducible and free. We use this result to prove that all cup-i constructions in the literature represent the same isomorphism class.
I don’t think we have such a separate article yet.
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