9 Categories

This chapter contains some elementary material about categories, functors, and natural transformations. Notably, we discuss and explore:

  1. Categories (Section 9.1).
  2. Examples of categories (Section 9.2).
  3. The quadruple adjunction $\pi _{0}\dashv {\webleft (-\webright )_{\mathsf{disc}}}\dashv \text{Obj}\dashv {\webleft (-\webright )_{\mathsf{indisc}}}$ between the category of categories and the category of sets (Section 9.3).
  4. Groupoids, categories in which all morphisms admit inverses (Section 9.4).
  5. Functors (Section 9.5).
  6. The conditions one may impose on functors in decreasing order of importance:
    1. Section 9.6 introduces the foundationally important conditions one may impose on functors, such as faithfulness, conservativity, essential surjectivity, etc.
    2. Section 9.7 introduces more conditions one may impose on functors that are still important but less omni-present than those of Section 9.6, such as being dominant, being a monomorphism, being pseudomonic, etc.
    3. Section 9.8 introduces some rather rare or uncommon conditions one may impose on functors that are nevertheless still useful to explicit record in this chapter.
  7. Natural transformations (Section 9.9).
  8. The various categorical and 2-categorical structures formed by categories, functors, and natural transformations (Section 9.10).
  • Section 9.1: Categories
    • Subsection 9.1.1: Foundations
      • Definition 9.1.1.1.1: Categories
      • Notation 9.1.1.1.2: Further Notation for Morphisms in Categories
      • Definition 9.1.1.1.3: Size Conditions on Categories
    • Subsection 9.1.2: Subcategories
    • Subsection 9.1.3: Skeletons of Categories
      • Definition 9.1.3.1.1: Skeletons of Categories
      • Definition 9.1.3.1.2: Skeletal Categories
      • Proposition 9.1.3.1.3: Properties of Skeletons of Categories
    • Subsection 9.1.4: Precomposition and Postcomposition
      • Definition 9.1.4.1.1: Precomposition and Postcomposition Functions
      • Proposition 9.1.4.1.2: Properties of Pre/Postcomposition
  • Section 9.2: Examples of Categories
    • Subsection 9.2.1: The Empty Category
    • Subsection 9.2.2: The Punctual Category
    • Subsection 9.2.3: Monoids as One-Object Categories
      • Example 9.2.3.1.1: Monoids as One-Object Categories
    • Subsection 9.2.4: Ordinal Categories
    • Subsection 9.2.5: The Walking Arrow
    • Subsection 9.2.6: More Examples of Categories
      • Example 9.2.6.1.1: More Examples of Categories
    • Subsection 9.2.7: Posetal Categories
      • Definition 9.2.7.1.1: Posetal Categories
      • Proposition 9.2.7.1.2: Properties of Posetal Categories
  • Section 9.3: The Quadruple Adjunction With Sets
    • Subsection 9.3.1: Statement
      • Proposition 9.3.1.1.1: The Quadruple Adjunction Between $\mathsf{Sets}$ and $\mathsf{Cats}$
    • Subsection 9.3.2: Connected Components and Connected Categories
      • Subsubsection 9.3.2.1: Connected Components of Categories
        • Definition 9.3.2.1.1: Connected Components of Categories
      • Subsubsection 9.3.2.2: Sets of Connected Components of Categories
        • Definition 9.3.2.2.1: Sets of Connected Components of Categories
        • Proposition 9.3.2.2.2: Properties of Sets of Connected Components
      • Subsubsection 9.3.2.3: Connected Categories
    • Subsection 9.3.3: Discrete Categories
      • Definition 9.3.3.1.1: Discrete Categories
      • Proposition 9.3.3.1.2: Properties of Discrete Categories on Sets
    • Subsection 9.3.4: Indiscrete Categories
      • Definition 9.3.4.1.1: Indiscrete Categories
      • Proposition 9.3.4.1.2: Properties of Indiscrete Categories on Sets
  • Section 9.4: Groupoids
    • Subsection 9.4.1: Isomorphisms
      • Definition 9.4.1.1.1: Isomorphisms
      • Notation 9.4.1.1.2: The Set of Isomorphisms Between Two Objects in a Category
    • Subsection 9.4.2: Groupoids
    • Subsection 9.4.3: The Groupoid Completion of a Category
      • Definition 9.4.3.1.1: The Groupoid Completion of a Category
      • Construction 9.4.3.1.2: Construction of the Groupoid Completion of a Category
      • Proposition 9.4.3.1.3: Properties of Groupoid Completion
    • Subsection 9.4.4: The Core of a Category
      • Definition 9.4.4.1.1: The Core of a Category
      • Notation 9.4.4.1.2: Alternative Notation for the Core of a Category
      • Construction 9.4.4.1.3: Construction of the Core of a Category
      • Proposition 9.4.4.1.4: Properties of the Core of a Category
  • Section 9.5: Functors
  • Section 9.6: Conditions on Functors
    • Subsection 9.6.1: Faithful Functors
      • Definition 9.6.1.1.1: Faithful Functors
      • Proposition 9.6.1.1.2: Properties of Faithful Functors
    • Subsection 9.6.2: Full Functors
      • Definition 9.6.2.1.1: Full Functors
      • Proposition 9.6.2.1.2: Properties of Full Functors
      • Question 9.6.2.1.3: Better Characterisations of Functors With Full Precomposition
    • Subsection 9.6.3: Fully Faithful Functors
      • Definition 9.6.3.1.1: Fully Faithful Functors
      • Proposition 9.6.3.1.2: Properties of Fully Faithful Functors
    • Subsection 9.6.4: Conservative Functors
      • Definition 9.6.4.1.1: Conservative Functors
      • Proposition 9.6.4.1.2: Properties of Conservative Functors
      • Question 9.6.4.1.3: Characterisations of Functors With Conservative Pre/Postcomposition
    • Subsection 9.6.5: Essentially Injective Functors
      • Definition 9.6.5.1.1: Essentially Injective Functors
      • Question 9.6.5.1.2: Characterisations of Functors With Essentially Injective Pre/Postcomposition
    • Subsection 9.6.6: Essentially Surjective Functors
      • Definition 9.6.6.1.1: Essentially Surjective Functors
      • Question 9.6.6.1.2: Characterisations of Functors With Essentially Surjective Pre/Postcomposition
    • Subsection 9.6.7: Equivalences of Categories
      • Definition 9.6.7.1.1: Equivalences of Categories
      • Proposition 9.6.7.1.2: Properties of Equivalences of Categories
    • Subsection 9.6.8: Isomorphisms of Categories
      • Definition 9.6.8.1.1: Isomorphisms of Categories
      • Example 9.6.8.1.2: Equivalent But Non-Isomorphic Categories
      • Proposition 9.6.8.1.3: Properties of Isomorphisms of Categories
  • Section 9.7: More Conditions on Functors
    • Subsection 9.7.1: Dominant Functors
      • Definition 9.7.1.1.1: Dominant Functors
      • Proposition 9.7.1.1.2: Properties of Dominant Functors
      • Question 9.7.1.1.3: Characterisations of Functors With Dominant Pre/Postcomposition
    • Subsection 9.7.2: Monomorphisms of Categories
      • Definition 9.7.2.1.1: Monomorphisms of Categories
      • Proposition 9.7.2.1.2: Properties of Monomorphisms of Categories
      • Question 9.7.2.1.3: Characterisations of Functors With Monic Pre/Postcomposition
    • Subsection 9.7.3: Epimorphisms of Categories
    • Subsection 9.7.4: Pseudomonic Functors
      • Definition 9.7.4.1.1: Pseudomonic Functors
      • Proposition 9.7.4.1.2: Properties of Pseudomonic Functors
    • Subsection 9.7.5: Pseudoepic Functors
      • Definition 9.7.5.1.1: Pseudoepic Functors
      • Proposition 9.7.5.1.2: Properties of Pseudoepic Functors
      • Question 9.7.5.1.3: Characterisations of Pseudoepic Functors
      • Question 9.7.5.1.4: Must a Pseudomonic and Pseudoepic Functor Be an Equivalence of Categories
      • Question 9.7.5.1.5: Characterisations of Functors With Pseudoepic Pre/Postcomposition
  • Section 9.8: Even More Conditions on Functors
    • Subsection 9.8.1: Injective on Objects Functors
      • Definition 9.8.1.1.1: Injective on Objects Functors
      • Proposition 9.8.1.1.2: Properties of Injective on Objects Functors
    • Subsection 9.8.2: Surjective on Objects Functors
      • Definition 9.8.2.1.1: Surjective on Objects Functors
    • Subsection 9.8.3: Bijective on Objects Functors
      • Definition 9.8.3.1.1: Bijective on Objects Functors
    • Subsection 9.8.4: Functors Representably Faithful on Cores
    • Subsection 9.8.5: Functors Representably Full on Cores
    • Subsection 9.8.6: Functors Representably Fully Faithful on Cores
    • Subsection 9.8.7: Functors Corepresentably Faithful on Cores
    • Subsection 9.8.8: Functors Corepresentably Full on Cores
    • Subsection 9.8.9: Functors Corepresentably Fully Faithful on Cores
  • Section 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
  • Section 9.10: Categories of Categories
    • Subsection 9.10.1: Functor Categories
      • Definition 9.10.1.1.1: Functor Categories
      • Proposition 9.10.1.1.2: Properties of Functor Categories
    • Subsection 9.10.2: The Category of Categories and Functors
      • Definition 9.10.2.1.1: The Category of Categories and Functors
      • Proposition 9.10.2.1.2: Properties of the Category $\mathsf{Cats}$
    • Subsection 9.10.3: The $2$-Category of Categories, Functors, and Natural Transformations
      • Definition 9.10.3.1.1: The $2$-Category of Categories
      • Proposition 9.10.3.1.2: Properties of the 2-Category $\mathsf{Cats}_{\mathsf{2}}$
    • Subsection 9.10.4: The Category of Groupoids
      • Definition 9.10.4.1.1: The Category of Small Groupoids
    • Subsection 9.10.5: The $2$-Category of Groupoids
      • Definition 9.10.5.1.1: The $2$-Category of Small Groupoids

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