Rikai Funo · Track 2 · middle
Ode to Category Theory
Abstract all mathematical structures into objects and morphisms — arrows between things. Functors map categories to categories; natural transformations map functors to functors. The unifying meta-language underneath all of modern mathematics, discovered by Eilenberg and Mac Lane in 1945 and still colonizing new territories of science. An ode to the arrow that erases what a thing IS and keeps only how it RELATES.
Lyrics
University of Chicago. Late in 1945. The stacks smell of binding glue and slow time. A sixty-four page block of serif type. There were too many things in the world. Topological spaces. Abelian groups. Each one a kingdom with its own rules. Samuel Eilenberg, you are thirty-two. Saunders Mac Lane, thirty-six. You sit in a room and you decide to forget. You decide the things themselves do not matter. All that matters is the arrow. The line I draw from A to B. The morphism. It erases the object. It keeps only the relation. You are not what you are. You are what you do. The diagrams were cut into zinc plates. Little squares that had to commute. Chasing the path around the block, always arriving at the same place. They called it abstract nonsense. But it wasn't nonsense. It was the only thing that made sense. To stop looking at the dots and only see the lines. All that matters is the arrow. The line I draw from A to B. My little morphism. It erases the object. It keeps only the relation. You are not what you are. You are what you do. And then a functor. An arrow between the worlds of arrows. Taking one whole category and mapping its ghost onto another. And then a natural transformation... An arrow between the arrows between the worlds. A general theory of natural equivalences. Not equality. Just... a way of translating that feels right. A structure-preserving map. And in the end, everything is gone. The groups, the spaces, the rings. All the substance melts away. Only you are left. The arrow. The path. The relation.