The Clowder Project

2.5 Characteristic Functions

• Subsection 2.5.1: The Characteristic Function of a Subset
  • Definition 2.5.1.1.1: The Characteristic Function of a Subset
  • Remark 2.5.1.1.2: Characteristic Functions of Subsets as Decategorifications of Presheaves
  • Notation 2.5.1.1.3: Further Notation for Characteristic Functions
  • Proposition 2.5.1.1.4: Properties of Characteristic Functions of Subsets
  • Remark 2.5.1.1.5: Powersets as Sets of Functions and Un/Straightening
• Subsection 2.5.2: The Characteristic Function of a Point
  • Definition 2.5.2.1.1: The Characteristic Function of a Point
  • Remark 2.5.2.1.2: Characteristic Functions of Points as Decategorifications of Representable Presheaves
• Subsection 2.5.3: The Characteristic Relation of a Set
  • Definition 2.5.3.1.1: The Characteristic Relation of a Set
  • Remark 2.5.3.1.2: The Characteristic Relation of a Set as a Decategorification of the Hom Profunctor
  • Proposition 2.5.3.1.3: Properties of Characteristic Relations
• Subsection 2.5.4: The Characteristic Embedding of a Set
  • Definition 2.5.4.1.1: The Characteristic Embedding of a Set
  • Remark 2.5.4.1.2: The Characteristic Embedding of a Set as a Decategorification of the Yoneda Embedding
  • Proposition 2.5.4.1.3: Properties of Characteristic Embeddings
• Subsection 2.5.5: The Yoneda Lemma for Sets
  • Proposition 2.5.5.1.1: The Yoneda Lemma for Sets
  • Corollary 2.5.5.1.2: The Characteristic Embedding Is Fully Faithful