The Clowder Project

8.5 Properties of the 2-Category of Relations

• Subsection 8.5.1: Self-Duality
  • Proposition 8.5.1.1.1: Self-Duality for the (2-)Category of Relations
• Subsection 8.5.2: Isomorphisms and Equivalences
  • Lemma 8.5.2.1.1: Conditions Involving a Relation and Its Converse I
  • Proposition 8.5.2.1.2: Isomorphisms and Equivalences in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.3: Internal Adjunctions
  • Proposition 8.5.3.1.1: Adjunctions in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.4: Internal Monads
  • Proposition 8.5.4.1.1: Internal Monads in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.4.1.2: Codensity Monads in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.5: Internal Comonads
  • Proposition 8.5.5.1.1: Internal Comonads in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.5.1.2: Density Comonads in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.6: Modules Over Internal Monads
  • Proposition 8.5.6.1.1: Modules Over Internal Monads in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.7: Comodules Over Internal Comonads
  • Proposition 8.5.7.1.1: Comodules Over Internal Comonads in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.8: Eilenberg–Moore and Kleisli Objects
  • Proposition 8.5.8.1.1: Eilenberg–Moore and Kleisli Objects in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.9: Co/Monoids
  • Remark 8.5.9.1.1: Co/Monoids in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.10: Monomorphisms and 2-Categorical Monomorphisms
  • Explanation 8.5.10.1.1: Monomorphisms in $\mathsf{Rel}$
  • Proposition 8.5.10.1.2: Characterisations of Monomorphisms in $\mathsf{Rel}$ I
  • Proposition 8.5.10.1.3: Characterisations of Monomorphisms in $\mathsf{Rel}$ II
  • Corollary 8.5.10.1.4: Characterisations of Monomorphisms in $\mathsf{Rel}$ III
  • Corollary 8.5.10.1.5: Characterisations of Monomorphisms in $\mathsf{Rel}$ V
  • Warning 8.5.10.1.6: Natural Conditions That Fail To Characterise Monomorphisms in $\mathsf{Rel}$
  • Remark 8.5.10.1.7: Monomorphisms in $\mathsf{Rel}$ Give Rise to Antichains
  • Proposition 8.5.10.1.8: Characterisations of 2-Categorical Monomorphisms in $\boldsymbol {\mathsf{Rel}}$ I
  • Proposition 8.5.10.1.9: Characterisations of 2-Categorical Monomorphisms in $\boldsymbol {\mathsf{Rel}}$ II
• Subsection 8.5.11: Epimorphisms and 2-Categorical Epimorphisms
  • Explanation 8.5.11.1.1: Epimorphisms in $\mathsf{Rel}$
  • Proposition 8.5.11.1.2: Characterisations of Epimorphisms in $\mathsf{Rel}$ I
  • Proposition 8.5.11.1.3: Characterisations of Epimorphisms in $\mathsf{Rel}$ II
  • Corollary 8.5.11.1.4: Characterisations of Epimorphisms in $\mathsf{Rel}$ III
  • Corollary 8.5.11.1.5: Characterisations of Epimorphisms in $\mathsf{Rel}$ IV
  • Corollary 8.5.11.1.6: Characterisations of Epimorphisms in $\mathsf{Rel}$ V
  • Warning 8.5.11.1.7: Natural Conditions That Fail To Characterise Epimorphisms in $\mathsf{Rel}$
  • Remark 8.5.11.1.8: Epimorphisms in $\mathsf{Rel}$ Give Rise to Antichains
  • Proposition 8.5.11.1.9: Characterisations of 2-Categorical Epimorphisms in $\boldsymbol {\mathsf{Rel}}$ I
  • Proposition 8.5.11.1.10: Characterisations of 2-Categorical Epimorphisms in $\boldsymbol {\mathsf{Rel}}$ II
• Subsection 8.5.12: Retractions
  • Proposition 8.5.12.1.1: Retractions in $\mathsf{Rel}$
• Subsection 8.5.13: Sections
  • Proposition 8.5.13.1.1: Sections in $\mathsf{Rel}$
• Subsection 8.5.14: Co/Limits
  • Proposition 8.5.14.1.1: Co/Limits in $\mathsf{Rel}$
• Subsection 8.5.15: Internal Left Kan Extensions
  • Proposition 8.5.15.1.1: Internal Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.15.1.2: Internal Left Kan Extensions Along Functions
  • Remark 8.5.15.1.3: Illustrating the Failure of Internal Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$ to Exist
  • Question 8.5.15.1.4: Existence of Specific Internal Left Kan Extensions of Relations
• Subsection 8.5.16: Internal Left Kan Lifts
  • Proposition 8.5.16.1.1: Internal Left Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.16.1.2: Internal Left Kan Lifts Along Functions
  • Question 8.5.16.1.3: Existence of Specific Internal Left Kan Lifts of Relations
• Subsection 8.5.17: Internal Right Kan Extensions
  • Motivation 8.5.17.1.1: Setting for Internal Right Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
  • Proposition 8.5.17.1.2: Internal Right Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.17.1.3: Examples of Internal Right Kan Extensions of Relations
  • Proposition 8.5.17.1.4: Properties of Internal Right Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.18: Internal Right Kan Lifts
  • Motivation 8.5.18.1.1: Setting for Internal Right Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
  • Proposition 8.5.18.1.2: Internal Right Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
  • Example 8.5.18.1.3: Examples of Internal Right Kan Extensions of Relations
  • Proposition 8.5.18.1.4: Properties of Internal Right Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.19: Closedness
  • Proposition 8.5.19.1.1: Closedness of $\boldsymbol {\mathsf{Rel}}$
• Subsection 8.5.20: $\mathsf{Rel}$ as a Category of Free Algebras
  • Proposition 8.5.20.1.1: $\mathsf{Rel}$ as a Category of Free Algebras