NOTES HONGKAI ZHU Contents 1. Set Theory 1.1. Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) 2. Category Theory 2.1. Category and Morphism Date: May 25, 2022. 1 2 2 2 2 2 HONGKAI ZHU 1. Set Theory 1.1. Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC). Axiom 1 (Axiom of Extensionality). Two sets are equal if they have the same elements. Axiom 2 (Axiom of Pairing). For every x, y, there exists a set x, y which contains x and y as elements. Axiom 3 (Axiom Schema of Specification (separation/restricted comprehension)). Suppose R is a property of sets, and P (u) represents a set u have the property P , then for every set X, there exists a set Y = {u ∈ X : P (u)}. ! Axiom 4 (Axiom of Union). For every set X, there exists a corresponding union X : {u : u ∈ v for some v ∈ X}. Axiom 5 (Axiom of Power Set). For every set X, its subset forms a set P (X) : {u : u ⊂ X}. Axiom 6 (Axiom of Infinity). There exists infinity sets. Axiom 7 (Axiom of Schema of Replacement). Suppose F is a function of X as domain, then there exists a set F (x) = {F (x) : x ∈ X}. Axiom 8 (Axiom of Regularity). Any nonempty set contains a membership ∈ minimal element. Axiom 9 (Axiom of ! Choice). Suppose every element in the set X is nonempty, then there exists a function g : X → X such that g(x) ∈ x for every x ∈ X. g is a choice function. 2. Category Theory 2.1. Category and Morphism. Definition 1. A category C is (i) A collection of objects Ob(C). (ii) To any two objects N, M ∈ Ob(C) there is attached a set HomC (N, M ) which is called the set of morphisms between these two objects. (iii) For any three objects N, M, P we have the composition of morphisms HomC (N, M ) × HomC (M, P ) → HomC (N, P ) (φ, ψ) %→ φ ◦ ψ. (iv) For any object N ∈ Ob(C) we have a distinguished element IdN ∈ HomC (N, N ), which is an identity on both sides under the composition. Usually we denote a morphism φ ∈ HomC (N, M ) by an arrow φ : N → M . If a morphism η is a composition of φ and ψ then we denote this by a commutative diagram (or commutative triangle) φ M N η ψ P