2 Constructions With Sets

This chapter develops some material relating to constructions with sets with an eye towards its categorical and higher-categorical counterparts to be introduced later in this work. Of particular interest are perhaps the following:

  1. Explicit descriptions of the major types of co/limits in $\mathsf{Sets}$, including in particular explicit descriptions of pushouts and coequalisers (see Definition 2.2.4.1.1, Remark 2.2.4.1.3, Definition 2.2.5.1.1, and Remark 2.2.5.1.3).
  2. A discussion of powersets as decategorifications of categories of presheaves, including in particular results such as:
    1. A discussion of the internal Hom of a powerset (Section 2.4.7).
    2. A 0-categorical version of the Yoneda lemma (, ), which we term the Yoneda lemma for sets (Proposition 2.5.5.1.1).
    3. A characterisation of powersets as free cocompletions (Section 2.4.5), mimicking the corresponding statement for categories of presheaves ().
    4. A characterisation of powersets as free completions (Section 2.4.6), mimicking the corresponding statement for categories of copresheaves ().
    5. A $\webleft (-1\webright )$-categorical version of un/straightening (Item 2 of Proposition 2.5.1.1.4 and Remark 2.5.1.1.5).
    6. A 0-categorical form of Isbell duality internal to powersets (Section 2.4.8).
  3. A lengthy discussion of the adjoint triple
    \[ f_{*}\dashv f^{-1}\dashv f_{!}\colon \mathcal{P}\webleft (A\webright )\overset {\rightleftarrows }{\to }\mathcal{P}\webleft (B\webright ) \]

    of functors (i.e. morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a map of sets $f\colon A\to B$, including in particular:

    1. How $f^{-1}$ can be described as a precomposition while $f_{*}$ and $f_{!}$ can be described as Kan extensions (Remark 2.6.1.1.3, Remark 2.6.2.1.2, and Remark 2.6.3.1.3).
    2. An extensive list of the properties of $f_{*}$, $f^{-1}$, and $f_{!}$ (Proposition 2.6.1.1.4, Proposition 2.6.1.1.5, Proposition 2.6.2.1.3, Proposition 2.6.2.1.4, Proposition 2.6.3.1.6, and Proposition 2.6.3.1.7).
    3. How the functors $f_{*}$, $f^{-1}$, $f_{!}$, along with the functors
      \begin{align*} -_{1}\cap -_{2} & \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ \webleft [-_{1},-_{2}\webright ]_{X} & \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}

      may be viewed as a six-functor formalism with the empty set $\text{Ø}$ as the dualising object (Section 2.6.4).

  • Section 2.1: Limits of Sets
  • Section 2.2: Colimits of Sets
    • Subsection 2.2.1: The Initial Set
      • Definition 2.2.1.1.1: The Initial Set
      • Construction 2.2.1.1.2: Construction of the Initial Set
    • Subsection 2.2.2: Coproducts of Families of Sets
      • Definition 2.2.2.1.1: The Coproduct of a Family of Sets
      • Construction 2.2.2.1.2: Construction of the Coproduct of a Family of Sets
      • Proposition 2.2.2.1.3: Properties of Coproducts of Families of Sets
    • Subsection 2.2.3: Binary Coproducts
      • Definition 2.2.3.1.1: Coproducts of Sets
      • Construction 2.2.3.1.2: Construction of Coproducts of Sets
      • Proposition 2.2.3.1.3: Properties of Coproducts of Sets
    • Subsection 2.2.4: Pushouts
    • Subsection 2.2.5: Coequalisers
    • Subsection 2.2.6: Direct Colimits
      • Definition 2.2.6.1.1: Direct Colimits of Sets
      • Construction 2.2.6.1.2: Construction of Direct Colimits of Sets
      • Lemma 2.2.6.1.3: Identification of $x$ with $f_{\alpha \beta }\webleft (x\webright )$ in Direct Colimits
      • Example 2.2.6.1.4: Examples of Direct Colimits of Sets
  • Section 2.3: Operations With Sets
    • Subsection 2.3.1: The Empty Set
    • Subsection 2.3.2: Singleton Sets
    • Subsection 2.3.3: Pairings of Sets
    • Subsection 2.3.4: Ordered Pairs
    • Subsection 2.3.5: Sets of Maps
    • Subsection 2.3.6: Unions of Families of Subsets
      • Definition 2.3.6.1.1: Unions of Families of Subsets
      • Proposition 2.3.6.1.2: Properties of Unions of Families of Subsets
    • Subsection 2.3.7: Intersections of Families of Subsets
      • Definition 2.3.7.1.1: Intersections of Families of Subsets
      • Proposition 2.3.7.1.2: Properties of Intersections of Families of Subsets
    • Subsection 2.3.8: Binary Unions
    • Subsection 2.3.9: Binary Intersections
      • Definition 2.3.9.1.1: Binary Intersections
      • Proposition 2.3.9.1.2: Properties of Binary Intersections
    • Subsection 2.3.10: Differences
    • Subsection 2.3.11: Complements
    • Subsection 2.3.12: Symmetric Differences
      • Definition 2.3.12.1.1: Symmetric Differences
      • Proposition 2.3.12.1.2: Properties of Symmetric Differences
  • Section 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
  • Section 2.5: Characteristic Functions
    • Subsection 2.5.1: The Characteristic Function of a Subset
      • Definition 2.5.1.1.1: The Characteristic Function of a Subset
      • Remark 2.5.1.1.2: Characteristic Functions of Subsets as Decategorifications of Presheaves
      • Notation 2.5.1.1.3: Further Notation for Characteristic Functions
      • Proposition 2.5.1.1.4: Properties of Characteristic Functions of Subsets
      • Remark 2.5.1.1.5: Powersets as Sets of Functions and Un/Straightening
    • Subsection 2.5.2: The Characteristic Function of a Point
      • Definition 2.5.2.1.1: The Characteristic Function of a Point
      • Remark 2.5.2.1.2: Characteristic Functions of Points as Decategorifications of Representable Presheaves
    • Subsection 2.5.3: The Characteristic Relation of a Set
      • Definition 2.5.3.1.1: The Characteristic Relation of a Set
      • Remark 2.5.3.1.2: The Characteristic Relation of a Set as a Decategorification of the Hom Profunctor
      • Proposition 2.5.3.1.3: Properties of Characteristic Relations
    • Subsection 2.5.4: The Characteristic Embedding of a Set
      • Definition 2.5.4.1.1: The Characteristic Embedding of a Set
      • Remark 2.5.4.1.2: The Characteristic Embedding of a Set as a Decategorification of the Yoneda Embedding
      • Proposition 2.5.4.1.3: Properties of Characteristic Embeddings
    • Subsection 2.5.5: The Yoneda Lemma for Sets
      • Proposition 2.5.5.1.1: The Yoneda Lemma for Sets
      • Corollary 2.5.5.1.2: The Characteristic Embedding Is Fully Faithful
  • Section 2.6: The Adjoint Triple $f_{*}\dashv f^{-1}\dashv f_{!}$
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