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 A(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 1ϕ is ϕ 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.