The Clowder Project

2.4 Powersets

• Subsection 2.4.1: Foundations
  • Definition 2.4.1.1.1: Powersets
  • Remark 2.4.1.1.2: Powersets as Decategorifications of Co/Presheaf Categories
  • Notation 2.4.1.1.3: Further Notation for Powersets
  • Proposition 2.4.1.1.4: Elementary Properties of Powersets
• Subsection 2.4.2: Functoriality of Powersets
  • Proposition 2.4.2.1.1: Functoriality of Powersets
• Subsection 2.4.3: Adjointness of Powersets I
  • Proposition 2.4.3.1.1: Adjointness of Powersets I
• Subsection 2.4.4: Adjointness of Powersets II
  • Proposition 2.4.4.1.1: Adjointness of Powersets II
• Subsection 2.4.5: Powersets as Free Cocompletions
  • Proposition 2.4.5.1.1: Powersets as Free Cocompletions: Universal Property
  • Proposition 2.4.5.1.2: Powersets as Free Cocompletions: Adjointness
  • Warning 2.4.5.1.3: Free Cocompletion Is Not an Idempotent Operation
• Subsection 2.4.6: Powersets as Free Completions
  • Proposition 2.4.6.1.1: Powersets as Free Completions: Universal Property
  • Proposition 2.4.6.1.2: Powersets as Free Completions: Adjointness
  • Warning 2.4.6.1.3: Free Completion Is Not an Idempotent Operation
• Subsection 2.4.7: The Internal Hom of a Powerset
  • Definition 2.4.7.1.1: The Internal Hom of a Powerset
  • Remark 2.4.7.1.2: Intuition for the Internal Hom of $\mathcal{P}\webleft (X\webright )$
  • Proposition 2.4.7.1.3: Properties of Internal Homs of Powersets
• Subsection 2.4.8: Isbell Duality for Sets
  • Definition 2.4.8.1.1: The Isbell Function
  • Remark 2.4.8.1.2: Motivation for the Isbell Function
  • Proposition 2.4.8.1.3: Isbell Duality for Sets