Category theory is a branch of mathematics that studies and analyzes different types of mapping between sets.
A category consists of a collection of objects, together with a collection of maps between those objects, called "morphisms", and a way to compose morphisms. Many familiar mathematical objects may be fit into a category:
- Set is the category of sets. The objects of Set are just sets, in the usual sense. A morphism between two objects A and B in set is just a function from A to B. The composition of morphisms is just the composition of set functions.
- Vect is the category of vector spaces over a field k. A morphism in this category is a linear transformation between two vector spaces.
- Top is the category of topological spaces, with maps as continuous functions.
Many concepts in mathematics, computer science and mathematical physics can be phrased in the language of categories and morphisms between categories. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 as a tool in the study of algebraic topology.
The power of category theory lies generally not in its power in producing new results, but as a unifying language showing that constructions in many different areas of mathematics are really analogous. An example of this is the product operation. Given two sets, it is possible to form their Cartesian product, which is simply the set of ordered pairs of elements from these sets. A similar construction is possible for topological spaces. Analogously, given two vector spaces one may construct a new vector space . These three objects all have many similar properties, and for many years these properties had to be explicitly proved each time such objects were dealt with in a new setting. Category theory gives a way to view all of these as examples of a single idea: the product of two objects is defined by a certain property of maps into that object. The crucial observation from category theory is that in any category, an object with the properties of "product" must satisfy various additional properties, which arise seemingly independently in all these examples.
Category theory is considered very powerful by some mathematicians. But is considered unnecessarily abstract by other mathematicians, so much so that they jokingly refer to it as "abstract nonsense".