Category Theory

Category theory was invented by Samuel Eilenberg and Saunders Mac Lane in the 1940s as a way of expressing certain constructions in algebraic topology. Category theory was developed rapidly in the subsequent decades. It has become an autonomous part of mathematics, studied for its own sake as well as being widely used as a unified language for the expression of mathematical ideas relating different fields.

For example, algebraic topology relates domains of interest in geometry to domains of interest in algebra. Algebraic geometry, on the other hand, goes in the opposite direction, associating, for example, with each commutative ring its spectrum of prime ideals. These fields were among the earliest to be studied using tools of category theory. Later applications came to abstract algebra, logic, computing science and physics, among others.

Why study categories

What are they good for? There is a range of answers for different backgrounds:

  • For mathematicians: category theory organises mathematical experience in a new and powerful way, revealing new connections and structure, and allows connections between diverse domains.
  • For computer scientists: category theory gives a precise handle on important notions such as types, compositionality, abstraction, representation-independence, genericity and more. It provides the fundamental mathematical structures underpinning many key computer science concepts.
  • For logicians: category theory gives a syntax-independent view of the fundamental structures of logic, and opens up new kinds of models and interpretations.
  • For philosophers: category theory opens up a fresh approach to structural foundations of mathematics and science and an alternative to the established focus on set theory.
  • For physicists: category theory offers new ways of formulating physical theories in a form that makes their structure explicit .

"... category theory is a theory which brings the notions of (type of) system and process to the forefront, two notions which are hard to cast within traditional monolithic mathematical structures."