Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that would generalize existing ones and provide a unified, more elegant and more efficient way to think of a class of objects.
What are your favorite examples of such generalizations and their authors ?
(NB: this is a soft question and largely an excuse to commemorate once more the passing of Alexander Grothendieck this week.)
Grothendieck, who is said to have been both very humble and at times very difficult to cope with, reportedly had (source, the quote comes from L. Schwartz’s biography) an argument with Jean Dieudonné who blamed him in his young years for “generalizing for the sole sake of generalizing”:
Dieudonné, avec l’agressivité (toujours passagère) dont il était capable, lui passa un savon mémorable, arguant qu’on ne devait pas travailler de cette manière, en généralisant pour le plaisir de généraliser.
Surely the step from numbers to groups and fields (which is due mostly to 19th-century mathematicians such as Abel, Galois and Dedekind) must count as one one of the greatest leaps forward in history.
Much of what was already known about numbers was quickly reproven for abstract algebraic structures – and thus for an infinite number of concrete structures with sheer limitless applications.