11 Categories

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

  1. 1.

    Categories (Section 11.1).

  2. 2.

    Examples of categories (Section 11.2).

  3. 3.

    The quadruple adjunction $\pi _{0}\dashv {\webleft (-\webright )_{\mathsf{disc}}}\dashv \operatorname {\mathrm{Obj}}\dashv {\webleft (-\webright )_{\mathsf{indisc}}}$ between the category of categories and the category of sets (Section 11.3).

  4. 4.

    Groupoids, categories in which all morphisms admit inverses (Section 11.4).

  5. 5.

    Functors (Section 11.5).

  6. 6.

    The conditions one may impose on functors in decreasing order of importance:

    1. (a)

      Section 11.6 introduces the foundationally important conditions one may impose on functors, such as faithfulness, conservativity, essential surjectivity, etc.

    2. (b)

      Section 11.7 introduces more conditions one may impose on functors that are still important but less omni-present than those of Section 11.6, such as being dominant, being a monomorphism, being pseudomonic, etc.

    3. (c)

      Section 11.8 introduces some rather rare or uncommon conditions one may impose on functors that are nevertheless still useful to explicit record in this chapter.

  7. 7.

    Natural transformations (Section 11.9).

  8. 8.

    The various categorical and 2-categorical structures formed by categories, functors, and natural transformations (Section 11.10).

  • Section 11.1: Categories
    • Subsection 11.1.1: Foundations
      • Definition 11.1.1.1.1: Categories
      • Notation 11.1.1.1.2: Further Notation for Morphisms in Categories
      • Definition 11.1.1.1.3: Size Conditions on Categories
    • Subsection 11.1.2: Subcategories
    • Subsection 11.1.3: Skeletons of Categories
      • Definition 11.1.3.1.1: Skeletons of Categories
      • Definition 11.1.3.1.2: Skeletal Categories
      • Proposition 11.1.3.1.3: Properties of Skeletons of Categories
    • Subsection 11.1.4: Precomposition and Postcomposition
      • Definition 11.1.4.1.1: Precomposition and Postcomposition Functions
      • Proposition 11.1.4.1.2: Properties of Pre/Postcomposition
  • Section 11.2: Examples of Categories
  • Section 11.3: The Quadruple Adjunction With Sets
    • Subsection 11.3.1: Statement
      • Proposition 11.3.1.1.1: The Quadruple Adjunction Between $\mathsf{Sets}$ and $\mathsf{Cats}$
    • Subsection 11.3.2: Connected Components and Connected Categories
      • Subsubsection 11.3.2.1: Connected Components of Categories
        • Definition 11.3.2.1.1: Connected Components of Categories
      • Subsubsection 11.3.2.2: Sets of Connected Components of Categories
        • Definition 11.3.2.2.1: Sets of Connected Components of Categories
        • Proposition 11.3.2.2.2: Properties of Sets of Connected Components
      • Subsubsection 11.3.2.3: Connected Categories
    • Subsection 11.3.3: Discrete Categories
      • Definition 11.3.3.1.1: Discrete Categories
      • Proposition 11.3.3.1.2: Properties of Discrete Categories on Sets
    • Subsection 11.3.4: Indiscrete Categories
      • Definition 11.3.4.1.1: Indiscrete Categories
      • Proposition 11.3.4.1.2: Properties of Indiscrete Categories on Sets
  • Section 11.4: Groupoids
    • Subsection 11.4.1: Isomorphisms
      • Definition 11.4.1.1.1: Isomorphisms
      • Notation 11.4.1.1.2: The Set of Isomorphisms Between Two Objects in a Category
    • Subsection 11.4.2: Groupoids
    • Subsection 11.4.3: The Groupoid Completion of a Category
      • Definition 11.4.3.1.1: The Groupoid Completion of a Category
      • Construction 11.4.3.1.2: Construction of the Groupoid Completion of a Category
      • Proposition 11.4.3.1.3: Properties of Groupoid Completion
    • Subsection 11.4.4: The Core of a Category
      • Definition 11.4.4.1.1: The Core of a Category
      • Notation 11.4.4.1.2: Alternative Notation for the Core of a Category
      • Construction 11.4.4.1.3: Construction of the Core of a Category
      • Proposition 11.4.4.1.4: Properties of the Core of a Category
  • Section 11.5: Functors
  • Section 11.6: Conditions on Functors
    • Subsection 11.6.1: Faithful Functors
    • Subsection 11.6.2: Full Functors
      • Definition 11.6.2.1.1: Full Functors
      • Proposition 11.6.2.1.2: Properties of Full Functors
      • Question 11.6.2.1.3: Better Characterisations of Functors With Full Precomposition
    • Subsection 11.6.3: Fully Faithful Functors
      • Definition 11.6.3.1.1: Fully Faithful Functors
      • Proposition 11.6.3.1.2: Properties of Fully Faithful Functors
    • Subsection 11.6.4: Conservative Functors
      • Definition 11.6.4.1.1: Conservative Functors
      • Proposition 11.6.4.1.2: Properties of Conservative Functors
      • Question 11.6.4.1.3: Characterisations of Functors With Conservative Pre/Postcomposition
    • Subsection 11.6.5: Essentially Injective Functors
      • Definition 11.6.5.1.1: Essentially Injective Functors
      • Question 11.6.5.1.2: Characterisations of Functors With Essentially Injective Pre/Postcomposition
    • Subsection 11.6.6: Essentially Surjective Functors
      • Definition 11.6.6.1.1: Essentially Surjective Functors
      • Question 11.6.6.1.2: Characterisations of Functors With Essentially Surjective Pre/Postcomposition
    • Subsection 11.6.7: Equivalences of Categories
      • Definition 11.6.7.1.1: Equivalences of Categories
      • Proposition 11.6.7.1.2: Properties of Equivalences of Categories
    • Subsection 11.6.8: Isomorphisms of Categories
      • Definition 11.6.8.1.1: Isomorphisms of Categories
      • Example 11.6.8.1.2: Equivalent But Non-Isomorphic Categories
      • Proposition 11.6.8.1.3: Properties of Isomorphisms of Categories
  • Section 11.7: More Conditions on Functors
    • Subsection 11.7.1: Dominant Functors
      • Definition 11.7.1.1.1: Dominant Functors
      • Proposition 11.7.1.1.2: Properties of Dominant Functors
      • Question 11.7.1.1.3: Characterisations of Functors With Dominant Pre/Postcomposition
    • Subsection 11.7.2: Monomorphisms of Categories
      • Definition 11.7.2.1.1: Monomorphisms of Categories
      • Proposition 11.7.2.1.2: Properties of Monomorphisms of Categories
      • Question 11.7.2.1.3: Characterisations of Functors With Monic Pre/Postcomposition
    • Subsection 11.7.3: Epimorphisms of Categories
      • Definition 11.7.3.1.1: Epimorphisms of Categories
      • Proposition 11.7.3.1.2: Properties of Epimorphisms of Categories
      • Question 11.7.3.1.3: Characterisations of Functors With Epic Pre/Postcomposition
    • Subsection 11.7.4: Pseudomonic Functors
      • Definition 11.7.4.1.1: Pseudomonic Functors
      • Proposition 11.7.4.1.2: Properties of Pseudomonic Functors
    • Subsection 11.7.5: Pseudoepic Functors
      • Definition 11.7.5.1.1: Pseudoepic Functors
      • Proposition 11.7.5.1.2: Properties of Pseudoepic Functors
      • Question 11.7.5.1.3: Characterisations of Pseudoepic Functors
      • Question 11.7.5.1.4: Must a Pseudomonic and Pseudoepic Functor Be an Equivalence of Categories
      • Question 11.7.5.1.5: Characterisations of Functors With Pseudoepic Pre/Postcomposition
  • Section 11.8: Even More Conditions on Functors
  • Section 11.9: Natural Transformations
    • Subsection 11.9.1: Transformations
      • Definition 11.9.1.1.1: Transformations
      • Notation 11.9.1.1.2: The Set of Transformations Between Two Functors
      • Remark 11.9.1.1.3: The Set of Transformations as a Product
    • Subsection 11.9.2: Natural Transformations
      • Definition 11.9.2.1.1: Natural Transformations
      • Remark 11.9.2.1.2: Further Terminology and Notation for Natural Transformations
      • Notation 11.9.2.1.3: The Set of Natural Transformations Between Two Functors
      • Definition 11.9.2.1.4: Equality of Natural Transformations
    • Subsection 11.9.3: Examples of Natural Transformations
      • Example 11.9.3.1.1: Identity Natural Transformations
      • Example 11.9.3.1.2: Natural Transformations Between Morphisms of Monoids
    • Subsection 11.9.4: Vertical Composition of Natural Transformations
      • Definition 11.9.4.1.1: Vertical Composition of Natural Transformations
      • Proposition 11.9.4.1.2: Properties of Vertical Composition of Natural Transformations
    • Subsection 11.9.5: Horizontal Composition of Natural Transformations
      • Definition 11.9.5.1.1: Horizontal Composition of Natural Transformations
      • Definition 11.9.5.1.2: Whiskering of Functors With Natural Transformations
      • Proposition 11.9.5.1.3: Properties of Horizontal Composition of Natural Transformations
    • Subsection 11.9.6: Properties of Natural Transformations
      • Proposition 11.9.6.1.1: Natural Transformations as Categorical Homotopies
    • Subsection 11.9.7: Natural Isomorphisms
      • Definition 11.9.7.1.1: Natural Isomorphisms
      • Proposition 11.9.7.1.2: Properties of Natural Isomorphisms
  • Section 11.10: Categories of Categories
    • Subsection 11.10.1: Functor Categories
    • Subsection 11.10.2: The Category of Categories and Functors
      • Definition 11.10.2.1.1: The Category of Categories and Functors
      • Proposition 11.10.2.1.2: Properties of the Category $\mathsf{Cats}$
    • Subsection 11.10.3: The $2$-Category of Categories, Functors, and Natural Transformations
      • Definition 11.10.3.1.1: The $2$-Category of Categories
      • Proposition 11.10.3.1.2: Properties of the 2-Category $\mathsf{Cats}_{\mathsf{2}}$
    • Subsection 11.10.4: The Category of Groupoids
      • Definition 11.10.4.1.1: The Category of Small Groupoids
    • Subsection 11.10.5: The $2$-Category of Groupoids
      • Definition 11.10.5.1.1: The $2$-Category of Small Groupoids
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