The Clowder Project

9.9 Natural Transformations

• Subsection 9.9.1: Transformations
  • Definition 9.9.1.1.1: Transformations
  • Notation 9.9.1.1.2: The Set of Transformations Between Two Functors
  • Remark 9.9.1.1.3: The Set of Transformations as a Product
• Subsection 9.9.2: Natural Transformations
  • Definition 9.9.2.1.1: Natural Transformations
  • Remark 9.9.2.1.2: Further Terminology and Notation for Natural Transformations
  • Notation 9.9.2.1.3: The Set of Natural Transformations Between Two Functors
  • Definition 9.9.2.1.4: Equality of Natural Transformations
• Subsection 9.9.3: Examples of Natural Transformations
  • Example 9.9.3.1.1: Identity Natural Transformations
  • Example 9.9.3.1.2: Natural Transformations Between Morphisms of Monoids
• Subsection 9.9.4: Vertical Composition of Natural Transformations
  • Definition 9.9.4.1.1: Vertical Composition of Natural Transformations
  • Proposition 9.9.4.1.2: Properties of Vertical Composition of Natural Transformations
• Subsection 9.9.5: Horizontal Composition of Natural Transformations
  • Definition 9.9.5.1.1: Horizontal Composition of Natural Transformations
  • Definition 9.9.5.1.2: Whiskering of Functors With Natural Transformations
  • Proposition 9.9.5.1.3: Properties of Horizontal Composition of Natural Transformations
• Subsection 9.9.6: Properties of Natural Transformations
  • Proposition 9.9.6.1.1: Natural Transformations as Categorical Homotopies
• Subsection 9.9.7: Natural Isomorphisms
  • Definition 9.9.7.1.1: Natural Isomorphisms
  • Proposition 9.9.7.1.2: Properties of Natural Isomorphisms