In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.

My source: Riehl

..."category" has been defined in order to be able to define "functor" and "functor" has been defined in order to be able to define "natural transformation."

My source: Mac Lane

In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.

My source: Riehl

... good general theory does not search for the maximum generality, but for the right generality.

My source: Mac Lane

... the subject languished until ... the discovery by Grothendieck that categories of sheaves (of abelian groups) over a topological space were abelian categories but not categories of modules, and that homological algebra in these categories was needed for a complete treatment of sheaf cohomology. With this impetus, abelian categories joined the establishment.

My source: Mac Lane