The concept of classicism is basically: classicism is a mathematical theory, which deals with the relationship between mathematical structures in an abstract way, treats mathematical concepts in an abstract way, and formalizes these concepts into groups of "objects" and "morphisms". Some people jokingly call it "generalized abstract nonsense", which appears in many branches of mathematics and some fields of theoretical computer science and mathematical physics.

background

The research category is to try to grasp the same characteristics of various related "mathematical structures" through axiomatic methods, and to link these structures through the "structure preservation function" between them. Therefore, the systematic study of category theory will allow any general conclusion of this mathematical structure to be proved by category axioms.

Consider the following example: a Grp class composed of groups contains all objects with a "group structure". In order to prove the theorem about groups, we can make logical deduction from this set of axioms. For example, from axioms, it can be immediately proved that the unit element of a group is unique.

Category theory will no longer only focus on single objects (such as groups) with specific structures, but on morphism (structure-preserving mapping) of these objects; By studying these morphisms, we can learn more about the structure of these objects. Take a group as an example, its homomorphism is group homomorphism.

Textbook learning

The most classic textbook of category theory is Mac Lane's The Category of Working Mathematicians.

But it needs a higher algebraic foundation, so some people on mathoverflow call it the category of working algebra. ?