mfe-mappinglisted

# Mapping ## Summary **Mapping** (Part VII: Mapping) Chapters: 22, 23, 24, 25 Plane Position: (0.2, 0.4) radius 0.4 Primitives: 42 Functions, categories, and information. How mathematical objects relate to each other — morphisms, entropy, signal processing. **Key Concepts:** Category, Probability Axioms, Functor, Natural Transformation, Shannon Entropy ## Key Primitives **Category** (definition): A category C consists of a collection of objects ob(C), a collection of morphisms hom(C) between objects, an identity morphism id_A for each object A, and a composition operation that is associative and respects identities. - Analyzing structure-preserving maps between mathematical objects - Identifying universal properties in algebraic structures - Abstracting common patterns across different areas of mathematics **Probability Axioms** (axiom): Kolmogorov's axioms: For a sample space Omega with sigma-algebra F, a probability measure P satisfies: (1) P(A) >= 0 for all A in F, (2) P(Omega) = 1, (3) P(union A_i) = sum P(A_i) for countably many disjoint events A_i. - Formalizing uncertainty in mathematical models - Defining the foundation for statistical inference - Setting up probability spaces for random experiments **Functor** (definition): A functor F: C -> D maps objects of C to objects of D and morphisms of C to morphisms of D, preserving identity morphisms F(id_A) = id_{F(A)} and composition F(g . f) = F(g) . F(f). - Translating problems between different