Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical meaning), where these collections satisfy some basic conditions.
For it is precisely what happened historically: category theory really started only after the discovery and use of Abelian categories, on the one hand and adjoint functiors, on the other hand. These discoveries were made approximately at the same time, namely in 1955 and 1956. It took another five to six years before the community could measure the extent of the changes these concepts brought about. It is approximately between 1963 and 1970 that category theory, as a genuine theory, arose and acquired a status that is still not clear to many mathematicians, logicians, and philosophers.
The applications of category theory to various scientific branches, starts from mathematics itself (algebraic topology, algebraic geometry), and passes over to theoretical computer science, music, quantum mechanics, biology, and psychology.
A branch of abstract, category theory was invented in the tradition of Felix Klein's Erlanger Programm, as a way of studying and characterizing different kinds of mathematical structures in terms of their "admissible transformations." The general notion of a category provides a characterization of the notion of a "structure-preserving transformation," and thereby of a species of structures admitting such transformations.
Categories originally arose in mathematics out of the need of a formalism to describe the passage from one type of mathematical structure to another. A category in this way represents a kind of mathematics, and may be described as category as mathematical workspace. A category is also a mathematical structure. As such, it is a common generalization of both ordered sets and monoids (the latter are a simple type of algebraic structure that include transition systems as examples), and questions motivated by those topics often have interesting answers for categories. This is category as mathematical structure.
Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic. It is a model of a collection of things with some structural similarity. It is a comparatively recent abstraction from the various abstract algebras developed in the early part of the 20th century.
The collaboration of Mac Lane with Samuel Eilenberg gave rise to the subject of category theory and their methods led to what came to be known later as homological algebra; both of these have proved highly influential in several branches of mathematics.
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
Category theory unifies mathematical structures in two different ways. First, as we have seen, almost every set theoretically defined mathematical structure with the appropriate notion of homomorphism yields a category. This is a unification provided within a set theoretical environment. Second, and perhaps even more important, once a type of structure has been defined, it is imperative to determine how new structures can be constructed out of the given one.
Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding and multiplying. At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated.
What is category theory? As a first approximation, one could say that category theory is the mathematical study of (abstract) algebras of functions. Just as group theory is the abstraction of the idea of a system of permutations of a set or symmetries of a geometric object, category theory arises from the idea of a system of functions among some objects.