Category Theory

Category Theory, as originally developed, is mathematics applied to the practice of mathematics itself. The concept of a category (not to be confused with the colloquial usage) models a particular mathematical discourse. Hence we have the category of sets, modeling set theory, the category of groups, modeling group theory, the category of vector spaces, modeling linear algebra, etc. This allows us to reason precisely about mathematics itself, and in particular about the relationships between different mathematical discourses, using the concepts of functor and natural transformation.

We had to discover the notion of a natural transformation. That in turn forced us to look at functors, which in turn made us look at categories.

—Saunders Mac Lane, in The Work of Samuel Eilenberg in Topology, 1976.

Applied Category Theory

If category theory is mathematics applied to the practice of mathematics, then it is already applied, by definition. So applied category theory is somewhat of a misnomer. What it really means is category theory applied outside of mathematics. The strength of this approach is that central concepts in mathematics, such as number, function, and equality are no longer taken for granted — instead, the meta-mathematics of category theory is utilized to create bespoke abstractions which perfectly fit the target domain, insofar as it can be described by thought-patterns that are stable enough that we can work with them collectively (see David Spivak's definition of math).