Category theory and structuralism

1. Category theory and structuralism Binghui, Shen

2. An introduction to Category theory • An introduction to Structuralism • Why • Further development

3. Definition of a category A category consists of the following data: • Objects: A,B, C, . . . • Arrows: f, g, h, . . . • For each arrow f, there are given objects dom(f), cod(f) called the domain and codomain of f. We write f : A → B to indicate that A = dom(f) and B = cod(f)

4. Definition of a category • Given arrows f : A → B and g : B → C, that is, with cod(f) = dom(g) there is given an arrow g ◦ f : A → C called the composite of f and g. • For each object A, there is given an arrow 1A : A → A called the identity arrow of A

5. Definition of a category • Associativity: h ◦ (g ◦ f) = (h ◦ g) ◦ f for all f : A → B, g : B → C, h : C → D. • Unit: f ◦ 1A = f = 1B ◦ f for all f : A → B.

6. Some Examples • Functions of sets: Let f be a function from set A to set B f : A → B. Now suppose we also have a function g : B → C, then there is a composite function g ◦ f : A → C, (g ◦ f)(a) = g(f(a)) a ∈ A.

7. Some Examples • Functions of sets: Now this operation “◦” of composition of functions is associative. Suppose: h : C → D

8. Some Examples • Functions of sets: we can compare (h ◦ g) ◦ f and h ◦ (g ◦ f) as indicated in the diagram given above. Two functions are always identical, (h ◦ g) ◦ f = h ◦ (g ◦ f) since for any a ∈ A, we have ((h ◦ g) ◦ f)(a) = h(g(f(a))) = (h ◦ (g ◦ f))(a)

9. Some Examples • Functions of sets: Finally, note that every set A has an identity function 1A : A → A 1A(a) = a.

10. Some Examples • A deductive system of logic there is an associated category of proofs, the objects are formulas: ϕ, ψ, . . . An arrow from ϕ to ψ is a deduction of ψ from the assumption ϕ.

11. Some Examples • A deductive system of logic Composition of arrows is given by putting together such deductions, which is associative. The identity arrows 1ϕisϕ to ϕ

12. STRUCTURALISM • The essence of a natural number is its relations to other natural numbers • a system is a collection of objects with certain relations among them. • structure is the abstract form of a system, highlighting the interrelationships among the objects • mathematics is the deductive study of structures as such.