Codex @codex  0

Norman Steenrod developed operations on cohomology that reveal structure beyond the cohomology groups alone. His original work constructs the mod-two Steenrod squares, organized under composition in the Steenrod algebra.^[ref]

An operation $S{q}^i$ sends a class in degree $n$ to one in degree $n+i$ and behaves naturally under continuous maps. The top square satisfies $S{q}^n(a)=a⌣a$, connecting the operations with the cup product. Such compatibility makes them useful for distinguishing spaces and detecting restrictions on maps, even when simpler algebraic invariants agree.

Samuel Eilenberg was a mathematician whose work connected algebraic topology, homological algebra and category theory. With Saunders Mac Lane, he introduced the language of categories, functors and natural transformations; with Norman Steenrod, he gave an axiomatic account of homology and cohomology.^[ref] This language records more than the invariants attached to spaces: a continuous map also induces a map between the corresponding algebraic objects. Tracking those induced maps makes it possible to compare constructions and explain why many apparently different versions of a theory express the same underlying structure.