4 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 4.2.4.1.1, Remark 4.2.4.1.3, Definition 4.2.5.1.1, and Remark 4.2.5.1.3).
-
2.
A discussion of powersets as decategorifications of categories of presheaves, including in particular results such as:
-
(a)
A discussion of the internal Hom of a powerset (Section 4.4.7).
-
(b)
A 0-categorical version of the Yoneda lemma (
, ), which we term the Yoneda lemma for sets (Proposition 4.5.5.1.1).
-
(c)
A characterisation of powersets as free cocompletions (Section 4.4.5), mimicking the corresponding statement for categories of presheaves (
).
-
(d)
A characterisation of powersets as free completions (Section 4.4.6), mimicking the corresponding statement for categories of copresheaves (
).
-
(e)
A $\webleft (-1\webright )$-categorical version of un/straightening (Item 2 of Proposition 4.5.1.1.4 and Remark 4.5.1.1.5).
-
(f)
A 0-categorical form of Isbell duality internal to powersets (Section 4.4.8).
-
(a)
-
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:
-
(a)
How $f^{-1}$ can be described as a precomposition while $f_{!}$ and $f_{*}$ can be described as Kan extensions (Remark 4.6.1.1.4, Remark 4.6.2.1.2, and Remark 4.6.3.1.4).
-
(b)
An extensive list of the properties of $f_{!}$, $f^{-1}$, and $f_{*}$ (Proposition 4.6.1.1.5, Proposition 4.6.1.1.6, Proposition 4.6.2.1.3, Proposition 4.6.2.1.4, Proposition 4.6.3.1.7, and Proposition 4.6.3.1.8).
-
(c)
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 4.6.4).
-
(a)
-
Section 4.1: Limits of Sets
- Subsection 4.1.1: The Terminal Set
-
Subsection 4.1.2: Products of Families of Sets
- Definition 4.1.2.1.1: The Product of a Family of Sets
- Construction 4.1.2.1.2: Construction of the Product of a Family of Sets
- Remark 4.1.2.1.3: Unwinding Construction 4.1.2.1.2
- Proposition 4.1.2.1.4: Properties of Products of Families of Sets
-
Subsection 4.1.3: Binary Products of Sets
- Definition 4.1.3.1.1: Binary Products of Sets
- Construction 4.1.3.1.2: Construction of Binary Products of Sets
- Proposition 4.1.3.1.3: Properties of Products of Sets
- Remark 4.1.3.1.4: The Cartesian Product of Sets as an $\webleft (\mathbb {E}_{k},\mathbb {E}_{\ell }\webright )$-Tensor Product
- Remark 4.1.3.1.5: Diagrams for Item 9, Item 10, Item 11, and Item 12 of Proposition 4.1.3.1.3
- Subsection 4.1.4: Pullbacks
- Subsection 4.1.5: Equalisers
- Subsection 4.1.6: Inverse Limits
-
Section 4.2: Colimits of Sets
- Subsection 4.2.1: The Initial Set
- Subsection 4.2.2: Coproducts of Families of Sets
- Subsection 4.2.3: Binary Coproducts
-
Subsection 4.2.4: Pushouts
- Definition 4.2.4.1.1: Pushouts of Sets
- Construction 4.2.4.1.2: Construction of Pushouts of Sets
- Remark 4.2.4.1.3: Unwinding Definition 4.2.4.1.1
- Remark 4.2.4.1.4: Pushouts of Sets Depend on the Maps
- Example 4.2.4.1.5: Examples of Pushouts of Sets
- Proposition 4.2.4.1.6: Properties of Pushouts of Sets
- Subsection 4.2.5: Coequalisers
- Subsection 4.2.6: Direct Colimits
-
Section 4.3: Operations With Sets
-
Subsection 4.3.1: The Empty Set
- Definition 4.3.1.1.1: The Empty Set
-
Subsection 4.3.2: Singleton Sets
- Definition 4.3.2.1.1: Singleton Sets
-
Subsection 4.3.3: Pairings of Sets
- Definition 4.3.3.1.1: Pairings of Sets
- Subsection 4.3.4: Ordered Pairs
- Subsection 4.3.5: Sets of Maps
- Subsection 4.3.6: Unions of Families of Subsets
- Subsection 4.3.7: Intersections of Families of Subsets
- Subsection 4.3.8: Binary Unions
- Subsection 4.3.9: Binary Intersections
-
Subsection 4.3.10: Differences
- Definition 4.3.10.1.1: Differences
- Proposition 4.3.10.1.2: Properties of Differences
-
Subsection 4.3.11: Complements
- Definition 4.3.11.1.1: Complements
- Proposition 4.3.11.1.2: Properties of Complements
-
Subsection 4.3.12: Symmetric Differences
- Definition 4.3.12.1.1: Symmetric Differences
- Proposition 4.3.12.1.2: Properties of Symmetric Differences
-
Subsection 4.3.1: The Empty Set
-
Section 4.4: Powersets
- Subsection 4.4.1: Foundations
-
Subsection 4.4.2: Functoriality of Powersets
- Proposition 4.4.2.1.1: Functoriality of Powersets
-
Subsection 4.4.3: Adjointness of Powersets I
- Proposition 4.4.3.1.1: Adjointness of Powersets I
-
Subsection 4.4.4: Adjointness of Powersets II
- Proposition 4.4.4.1.1: Adjointness of Powersets II
- Subsection 4.4.5: Powersets as Free Cocompletions
- Subsection 4.4.6: Powersets as Free Completions
- Subsection 4.4.7: The Internal Hom of a Powerset
- Subsection 4.4.8: Isbell Duality for Sets
-
Section 4.5: Characteristic Functions
-
Subsection 4.5.1: The Characteristic Function of a Subset
- Definition 4.5.1.1.1: The Characteristic Function of a Subset
- Remark 4.5.1.1.2: Characteristic Functions of Subsets as Decategorifications of Presheaves
- Notation 4.5.1.1.3: Further Notation for Characteristic Functions
- Proposition 4.5.1.1.4: Properties of Characteristic Functions of Subsets
- Remark 4.5.1.1.5: Powersets as Sets of Functions and Un/Straightening
- Subsection 4.5.2: The Characteristic Function of a Point
- Subsection 4.5.3: The Characteristic Relation of a Set
- Subsection 4.5.4: The Characteristic Embedding of a Set
- Subsection 4.5.5: The Yoneda Lemma for Sets
-
Subsection 4.5.1: The Characteristic Function of a Subset
-
Section 4.6: The Adjoint Triple $f_{!}\dashv f^{-1}\dashv f_{*}$
-
Subsection 4.6.1: Direct Images
- Definition 4.6.1.1.1: Direct Images
- Notation 4.6.1.1.2: Further Notation for Direct Images
- Warning 4.6.1.1.3: Notation for Direct Images Is Confusing
- Remark 4.6.1.1.4: Unwinding Definition 4.6.1.1.1
- Proposition 4.6.1.1.5: Properties of Direct Images I
- Proposition 4.6.1.1.6: Properties of Direct Images II
-
Subsection 4.6.2: Inverse Images
- Definition 4.6.2.1.1: Inverse Images
- Remark 4.6.2.1.2: Unwinding Definition 4.6.2.1.1
- Proposition 4.6.2.1.3: Properties of Inverse Images I
- Proposition 4.6.2.1.4: Properties of Inverse Images II
-
Subsection 4.6.3: Codirect Images
- Definition 4.6.3.1.1: Codirect Images
- Notation 4.6.3.1.2: Further Notation for Codirect Images
- Warning 4.6.3.1.3: Notation for Codirect Images Is Confusing
- Remark 4.6.3.1.4: Unwinding Definition 4.6.3.1.1
- Definition 4.6.3.1.5: The Image and Complement Parts of $f_{*}$
- Example 4.6.3.1.6: Examples of Codirect Images
- Proposition 4.6.3.1.7: Properties of Codirect Images I
- Proposition 4.6.3.1.8: Properties of Codirect Images II
- Subsection 4.6.4: A Six-Functor Formalism for Sets
-
Subsection 4.6.1: Direct Images
