9 Types of Morphisms in Bicategories

In this chapter, we study special kinds of morphisms in bicategories:

1. Monomorphisms and Epimorphisms in Bicategories (Section 9.1 and Section 9.2). There is a large number of different notions capturing the idea of a “monomorphism” or of an “epimorphism” in a bicategory.

   Arguably, the notion that best captures these concepts is that of a pseudomonic morphism (Definition 9.1.10.1.1) and of a pseudoepic morphism (Definition 9.2.10.1.1), although the other notions introduced in Section 9.1 and Section 9.2 are also interesting on their own.

• Section 9.1: Monomorphisms in Bicategories
  • Subsection 9.1.1: Representably Faithful Morphisms
    • Definition 9.1.1.1.1: Representably Faithful Morphisms
    • Remark 9.1.1.1.2: Unwinding Definition 9.1.1.1.1
    • Example 9.1.1.1.3: Examples of Representably Faithful Morphisms
  • Subsection 9.1.2: Representably Full Morphisms
    • Definition 9.1.2.1.1: Representably Full Morphisms
    • Remark 9.1.2.1.2: Unwinding Definition 9.1.2.1.1
    • Example 9.1.2.1.3: Examples of Representably Full Morphisms
  • Subsection 9.1.3: Representably Fully Faithful Morphisms
    • Definition 9.1.3.1.1: Representably Fully Faithful Morphisms
    • Remark 9.1.3.1.2: Unwinding Representably Fully Faithful Morphisms
    • Example 9.1.3.1.3: Examples of Representably Fully Faithful Morphisms
  • Subsection 9.1.4: Morphisms Representably Faithful on Cores
    • Definition 9.1.4.1.1: Morphisms Representably Faithful on Cores
    • Remark 9.1.4.1.2: Unwinding Definition 9.1.4.1.1
  • Subsection 9.1.5: Morphisms Representably Full on Cores
    • Definition 9.1.5.1.1: Morphisms Representably Full on Cores
    • Remark 9.1.5.1.2: Unwinding Definition 9.1.5.1.1
  • Subsection 9.1.6: Morphisms Representably Fully Faithful on Cores
    • Definition 9.1.6.1.1: Morphisms Representably Fully Faithful on Cores
    • Remark 9.1.6.1.2: Unwinding Definition 9.1.6.1.1
  • Subsection 9.1.7: Representably Essentially Injective Morphisms
    • Definition 9.1.7.1.1: Representably Essentially Injective Morphisms
    • Remark 9.1.7.1.2: Unwinding Definition 9.1.7.1.1
  • Subsection 9.1.8: Representably Conservative Morphisms
    • Definition 9.1.8.1.1: Representably Conservative Morphisms
    • Remark 9.1.8.1.2: Unwinding Definition 9.1.8.1.1
  • Subsection 9.1.9: Strict Monomorphisms
    • Definition 9.1.9.1.1: Strict Monomorphisms
    • Remark 9.1.9.1.2: Unwinding Definition 9.1.9.1.1
    • Example 9.1.9.1.3: Examples of Strict Monomorphisms
  • Subsection 9.1.10: Pseudomonic Morphisms
    • Definition 9.1.10.1.1: Pseudomonic Morphisms
    • Remark 9.1.10.1.2: Unwinding Definition 9.1.10.1.1
    • Proposition 9.1.10.1.3: Properties of Pseudomonic Morphisms
• Section 9.2: Epimorphisms in Bicategories
  • Subsection 9.2.1: Corepresentably Faithful Morphisms
    • Definition 9.2.1.1.1: Corepresentably Faithful Morphisms
    • Remark 9.2.1.1.2: Unwinding Definition 9.2.1.1.1
    • Example 9.2.1.1.3: Examples of Corepresentably Faithful Morphisms
  • Subsection 9.2.2: Corepresentably Full Morphisms
    • Definition 9.2.2.1.1: Corepresentably Full Morphisms
    • Remark 9.2.2.1.2: Unwinding Definition 9.2.2.1.1
    • Example 9.2.2.1.3: Examples of Corepresentably Full Morphisms
  • Subsection 9.2.3: Corepresentably Fully Faithful Morphisms
    • Definition 9.2.3.1.1: Corepresentably Fully Faithful Morphisms
    • Remark 9.2.3.1.2: Unwinding Definition 9.2.3.1.1
    • Example 9.2.3.1.3: Examples of Corepresentably Fully Faithful Morphisms
  • Subsection 9.2.4: Morphisms Corepresentably Faithful on Cores
    • Definition 9.2.4.1.1: Morphisms Corepresentably Faithful on Cores
    • Remark 9.2.4.1.2: Unwinding Definition 9.2.4.1.1
  • Subsection 9.2.5: Morphisms Corepresentably Full on Cores
    • Definition 9.2.5.1.1: Morphisms Corepresentably Full on Cores
    • Remark 9.2.5.1.2: Unwinding Definition 9.2.5.1.1
  • Subsection 9.2.6: Morphisms Corepresentably Fully Faithful on Cores
    • Definition 9.2.6.1.1: Morphisms Corepresentably Fully Faithful on Cores
    • Remark 9.2.6.1.2: Unwinding Definition 9.2.6.1.1
  • Subsection 9.2.7: Corepresentably Essentially Injective Morphisms
    • Definition 9.2.7.1.1: Corepresentably Essentially Injective Morphisms
    • Remark 9.2.7.1.2: Unwinding Definition 9.2.7.1.1
  • Subsection 9.2.8: Corepresentably Conservative Morphisms
    • Definition 9.2.8.1.1: Corepresentably Conservative Morphisms
    • Remark 9.2.8.1.2: Unwinding Definition 9.2.8.1.1
  • Subsection 9.2.9: Strict Epimorphisms
    • Definition 9.2.9.1.1: Strict Epimorphisms
    • Remark 9.2.9.1.2: Unwinding Definition 9.2.9.1.1
    • Example 9.2.9.1.3: Examples of Strict Epimorphisms
  • Subsection 9.2.10: Pseudoepic Morphisms
    • Definition 9.2.10.1.1: Pseudoepic Morphisms
    • Remark 9.2.10.1.2: Unwinding Definition 9.2.10.1.1
    • Proposition 9.2.10.1.3: Properties of Pseudoepic Morphisms