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:
-
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).
-
A discussion of powersets as decategorifications of categories of presheaves, including in particular results such as:
-
A discussion of the internal Hom of a powerset (Section 2.4.7).
-
A 0-categorical version of the Yoneda lemma (
, ), which we term the Yoneda lemma for sets (Proposition 2.5.5.1.1).
-
A characterisation of powersets as free cocompletions (Section 2.4.5), mimicking the corresponding statement for categories of presheaves ().
-
A characterisation of powersets as free completions (Section 2.4.6), mimicking the corresponding statement for categories of copresheaves ().
-
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).
-
A 0-categorical form of Isbell duality internal to powersets (Section 2.4.8).
-
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:
-
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).
-
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).
-
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
-
Subsection 2.1.1: The Terminal Set
-
Definition 2.1.1.1.1: The Terminal Set
-
Construction 2.1.1.1.2: Construction of the Terminal Set
-
Subsection 2.1.2: Products of Families of Sets
-
Subsection 2.1.3: Binary Products of Sets
-
Subsection 2.1.4: Pullbacks
-
Definition 2.1.4.1.1: Pullbacks of Sets
-
Construction 2.1.4.1.2: Construction of Pullbacks of Sets
-
Example 2.1.4.1.4: Examples of Pullbacks of Sets
-
Proposition 2.1.4.1.5: Properties of Pullbacks of Sets
-
Subsection 2.1.5: Equalisers
-
Definition 2.1.5.1.1: Equalisers of Sets
-
Construction 2.1.5.1.2: Construction of Equalisers of Sets
-
Proposition 2.1.5.1.3: Properties of Equalisers of Sets
-
Subsection 2.1.6: Inverse Limits
-
Definition 2.1.6.1.1: Inverse Limits of Sets
-
Construction 2.1.6.1.2: Construction of Inverse Limits of Sets
-
Example 2.1.6.1.3: Examples of Inverse 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 $f_{*}\dashv f^{-1}\dashv f_{!}$
-
Subsection 2.6.1: Direct Images
-
Subsection 2.6.2: Inverse Images
-
Subsection 2.6.3: Direct Images With Compact Support
-
Definition 2.6.3.1.1: Direct Images With Compact Support
-
Notation 2.6.3.1.2: Further Notation for Direct Images With Compact Support
-
Remark 2.6.3.1.3: Unwinding Definition 2.6.3.1.1
-
Definition 2.6.3.1.4: The Image and Complement Parts of $f_{!}$
-
Example 2.6.3.1.5: Examples of Direct Images With Compact Support
-
Proposition 2.6.3.1.6: Properties of Direct Images With Compact Support I
-
Proposition 2.6.3.1.7: Properties of Direct Images With Compact Support II
-
Subsection 2.6.4: A Six-Functor Formalism for Sets
-
Remark 2.6.4.1.1: A Six-Functor Formalism for Sets
-
Proposition 2.6.4.1.2: A Six Functor Formalism for Sets