Algebraic topology is the discipline that studies geometries by associating them with algebraic objects (usually, groups or vector spaces) and observing how changing the underlying space affects the related algebras. In 1941, two mathematicians working in that field sought to generalize a theorem that they discovered, and needed to show that their solution was still valid for a larger class of spaces, obtained by “natural” transformations. Natural, at that point, was a term lacking a precise definition, and only meant something like “avoiding arbitrary choices”, in the same way a vector space is naturally isomorphic to its double dual, while it’s isomorphic to its dual only through the choice of a basis.
The need to make precise the notion of naturality for algebraic topology led them to the definition of natural transformation, which in turn required the notion of functor which in turn required the notion of category.
This answers questions 1 and 2: category theory was born to give a precise definition of naturality, and was sought to generalize the “universal coefficient theorem” to a larger class of spaces.
This story is told with a lot of details in the first paragraphs of Riehl’s wonderful “Category theory in context”.
To answer n° 3, though, even if category theory was rapidly expanding during the ’50s and the ‘60s, it was only with the work of Lawvere (who I consider a genius on par with Gödel) in the ’70s that it became a foundational discipline: guided by his intuitions, category theory became the unifying language for every branch of mathematics, from geometry to computation to logic to algebras. Basically, it showed how the variety of mathematical disciplines are just different ways to say the same thing.
Algebraic topology is the discipline that studies geometries by associating them with algebraic objects (usually, groups or vector spaces) and observing how changing the underlying space affects the related algebras. In 1941, two mathematicians working in that field sought to generalize a theorem that they discovered, and needed to show that their solution was still valid for a larger class of spaces, obtained by “natural” transformations. Natural, at that point, was a term lacking a precise definition, and only meant something like “avoiding arbitrary choices”, in the same way a vector space is naturally isomorphic to its double dual, while it’s isomorphic to its dual only through the choice of a basis.
The need to make precise the notion of naturality for algebraic topology led them to the definition of natural transformation, which in turn required the notion of functor which in turn required the notion of category.
This answers questions 1 and 2: category theory was born to give a precise definition of naturality, and was sought to generalize the “universal coefficient theorem” to a larger class of spaces.
This story is told with a lot of details in the first paragraphs of Riehl’s wonderful “Category theory in context”.
To answer n° 3, though, even if category theory was rapidly expanding during the ’50s and the ‘60s, it was only with the work of Lawvere (who I consider a genius on par with Gödel) in the ’70s that it became a foundational discipline: guided by his intuitions, category theory became the unifying language for every branch of mathematics, from geometry to computation to logic to algebras. Basically, it showed how the variety of mathematical disciplines are just different ways to say the same thing.