Maths in a Minute: Category Theory

Category theory was first developed in the 1940s by pure mathematicians Samuel Eilenberg and Saunders Mac Lane. A category consists of a family of abstract objects and morphisms between pairs of objects. You might imagine the objects as blobs and morphisms as arrows that point from one blob to another blob. They indicate some sort of relationship between the corresponding objects. Finally there's a rule for how two arrows can combine (see here for the details).

The definition appears so abstract as to be meaningless, but here's a simple example of a category: think of the natural numbers 1, 2, 3, 4, …. Draw an arrow from a number n to a number m if n < m. There's an obvious way of combining two arrows. If n < m < l  then n < l, so the arrow from n to m combined with the arrow from m to l gives an arrow from n to l.

This example captures how category theory applies to something as ordinary as the number line, but it also has applications outside of maths. Find out more in Category theory: From genes to gravity and From abstract nonsense to essential tool.