What Are the Fundamental Axioms of Category Theory?

[MathematicalPhysicist]

what are them?

 

[MathematicalPhysicist]

never mind i found them in here:http://plato.stanford.edu/entries/category-theory/

 

[tacman]

The axioms of category theory are a set of fundamental principles that define the structure and behavior of categories. These axioms serve as the foundation for the study of category theory, which is a branch of mathematics that focuses on the abstract structure and relationships between mathematical objects.

The first axiom of category theory is the identity axiom, which states that every object in a category must have at least one identity morphism, a special type of function that maps an object to itself. This axiom ensures that each object in a category can be uniquely identified and that there is a consistent way to compose morphisms.

The second axiom is the composition axiom, which states that for any three morphisms in a category, if the first two can be composed, then the result of that composition can also be composed with the third. This axiom ensures that the composition of morphisms is always well-defined and that the resulting morphism is also a member of the same category.

The final axiom is the associativity axiom, which states that the order in which morphisms are composed does not matter, as long as the composition is valid. This axiom ensures that the composition of morphisms is a consistent and associative operation, similar to how addition is associative in arithmetic.

Together, these axioms provide a framework for understanding the structure and behavior of categories and the relationships between objects and morphisms within them. They also allow for the development of more complex concepts and theorems in category theory, making it a powerful tool for studying a wide range of mathematical structures.