Category Theory book

Table of contents

• Numbers
  • Integers
    • Is multiplication repeated addition
  • Rational numbers
  • Real numbers
  • Complex numbers
    • Gaussian integers
    • Eisenstein integers
  • Quaternions, etc
    • Hurwitz quaternions
  • p-adics
    • What's the square root of -1 in the 5-adics?
  • Matrices
  • Polynomials
  • Linear equations
• Arithmetic
  • Peano arithmetic
  • Divisors, primes and coprimes
  • Fundamental theorem of arithmetic
  • Chinese remainder theorem (and Nicomachus)
    • Things are easier under modular arithmetic
• Set theory
  • Naive set theory
  • Foundational issues and paradoxes
  • Zermelo-Fraenkel
  • Infinities
• Graph theory
  • Tarjan's algorithm and "dividing out the cycles"
  • DAGs, Lamport timestamps and Git
• Linear algebra
  • Matrices are really linear transformations (not just two-dimensional arrays)
  • Quantum superpositions are really only linear combinations
• Abelian groups
  • Generators and representations
  • Cyclic groups
  • Fundamental theorem of finite commutative groups
• Groups
  • Dihedral groups
  • Symmetric groups
  • Alternating groups
  • Rubik's cube
  • Generalizing: monoids, semigroups, magmas
• Group theory
  • Group homomorphisms
  • Monomorphisms and epimorphisms
  • Direct product
  • Semidirect product
    • Dihedral groups
    • Translations and rotations
    • Explaining the structure of the Rubik's cube group
  • The first isomorphism theorem
  • Galois and solvability
• Geometry
  • Conic sections
  • Symmetries on a sphere
  • The platonic solids
  • The 4D versions of these
  • Archimedean solids
  • Johnson solids
  • Goldberg polyhedra
  • Tilings of the plane
  • Spherical, planar, hyperbolic geometry
  • Octagonal tiling
  • Triangle groups, Coxeter groups and Coxeter-Dynkin digrams
• Field theory (?)
• Topology
  • Jordan's curve theorem
  • Classification of closed 2-dimensional manifolds
    • An algorithm to create the manifolds by gluing together polygons
    • ZIP proof
  • Fundamental groups
  • Homology
• Logic and foundations
  • Gödel and undecidability
  • Intuitionistic logic
    • Martin-Löf type theory and Agda
  • Linear logic
• Category theory, finally
  • Composition
  • Rebuilding math using categories
  • Functors
  • Products and coproducts
  • Duality
  • Limits
  • Universal properties
  • Adjunctions
  • Natural transformations
  • The Yoneda Lemma
  • Enriched categories, 2-categories, and n-categories
  • Foundations and category theory