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. In particular, it contains:

  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 2.2.4.1.2, Definition 2.2.5.1.1, and Remark 2.2.5.1.2).
  2. A discussion of powersets as decategorifications of categories of presheaves (Remark 2.4.1.1.2 and Remark 2.4.3.1.2), including a $\webleft (-1\webright )$-categorical analogue of un/straightening, described in Item 1 and Item 2 of Proposition 2.4.3.1.6 and Remark 2.4.3.1.7.
  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 (morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a map of sets $f\colon A\to B$, along with a discussion of the properties of $f_{*}$, $f^{-1}$, and $f_{!}$.

    In line with the categorical viewpoint developed here, this adjoint triple may be described in terms of Kan extensions, and, as it turns out, it also shows up in some definitions and results in point-set topology, such as in e.g. notions of continuity for functions ().

  • Section 2.1: Limits of Sets
    • Subsection 2.1.1: The Terminal Set
    • Subsection 2.1.2: Products of Families of Sets
      • Definition 2.1.2.1.1: The Product of a Family of Sets
      • Proposition 2.1.2.1.2: Properties of Products of Families of Sets
    • Subsection 2.1.3: Binary Products of Sets
      • Definition 2.1.3.1.1: Products of Sets
      • Proposition 2.1.3.1.2: Properties of Products of Sets
    • Subsection 2.1.4: Pullbacks
      • Definition 2.1.4.1.1: Pullbacks of Sets
      • Example 2.1.4.1.2: Examples of Pullbacks of Sets
      • Proposition 2.1.4.1.3: Properties of Pullbacks of Sets
    • Subsection 2.1.5: Equalisers
      • Definition 2.1.5.1.1: Equalisers of Sets
      • Proposition 2.1.5.1.2: Properties of Equalisers of Sets
  • Section 2.2: 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
    • Subsection 2.3.7: Binary Unions
    • Subsection 2.3.8: Intersections of Families
      • Definition 2.3.8.1.1: Intersections of Families
    • Subsection 2.3.9: Binary Intersections
      • Definition 2.3.9.1.1: Binary Intersections
      • Proposition 2.3.9.1.2: Properties of Binary Intersections
      • Remark 2.3.9.1.3: Intuition for the Internal $\mathbf{Hom}$ of $\mathcal{P}\webleft (X\webright )$
    • 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: Characteristic Functions
      • Definition 2.4.1.1.1: Characteristic Functions
      • Remark 2.4.1.1.2: Characteristic Functions as Decategorifications of Presheaves
      • Proposition 2.4.1.1.3: Properties of Characteristic Functions
    • Subsection 2.4.2: The Yoneda Lemma for Sets
      • Proposition 2.4.2.1.1: The Yoneda Lemma for Sets
      • Corollary 2.4.2.1.2: The Characteristic Embedding Is Fully Faithful
    • Subsection 2.4.3: Powersets
      • Definition 2.4.3.1.1: Powersets
      • Remark 2.4.3.1.2: Powersets as Decategorifications of Co/Presheaf Categories
      • Proposition 2.4.3.1.3: Properties of Powersets: As Categories
      • Proposition 2.4.3.1.4: Properties of Powersets: Functoriality and Adjointness
      • Proposition 2.4.3.1.5: Properties of Powersets: Monoidality
      • Proposition 2.4.3.1.6: Properties of Powersets: As Sets of Functions/Relations
      • Remark 2.4.3.1.7: Powersets as Sets of Functions and Un/Straightening
      • Proposition 2.4.3.1.8: Properties of Powersets: As Free Cocompletions
    • Subsection 2.4.4: Direct Images
    • Subsection 2.4.5: Inverse Images
    • Subsection 2.4.6: Direct Images With Compact Support
      • Definition 2.4.6.1.1: Direct Images With Compact Support
      • Notation 2.4.6.1.2: Further Notation for Direct Images With Compact Support
      • Remark 2.4.6.1.3: Unwinding Definition 2.4.6.1.1
      • Definition 2.4.6.1.4: The Image and Complement Parts of $f_{!}$
      • Example 2.4.6.1.5: Examples of Direct Images With Compact Support
      • Proposition 2.4.6.1.6: Properties of Direct Images With Compact Support I
      • Proposition 2.4.6.1.7: Properties of Direct Images With Compact Support II

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