An axiomatic characterization of Steenrod's cup-$i$ products

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.

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