4 Pointed Sets

This chapter contains some foundational material on pointed sets.

• Section 4.1: Pointed Sets
  • Subsection 4.1.1: Foundations
    • Definition 4.1.1.1.1: Pointed Sets
    • Remark 4.1.1.1.2: Unwinding Definition 4.1.1.1.1
    • Example 4.1.1.1.3: The Zero Sphere
    • Example 4.1.1.1.4: The Trivial Pointed Set
    • Example 4.1.1.1.5: The Standard Pointed Set With $n+1$ Elements
  • Subsection 4.1.2: Morphisms of Pointed Sets
    • Definition 4.1.2.1.1: Morphisms of Pointed Sets
    • Remark 4.1.2.1.2: Unwinding Definition 4.1.2.1.1
  • Subsection 4.1.3: The Category of Pointed Sets
    • Definition 4.1.3.1.1: The Category of Pointed Sets
    • Remark 4.1.3.1.2: Unwinding Definition 4.1.3.1.1
  • Subsection 4.1.4: Elementary Properties of Pointed Sets
    • Proposition 4.1.4.1.1: Elementary Properties of Pointed Sets
  • Subsection 4.1.5: Active and Inert Morphisms of Pointed Sets
    • Definition 4.1.5.1.1: Active and Inert Morphisms of Pointed Sets
    • Notation 4.1.5.1.2: The Category of Pointed Sets and Active Morphisms
    • Example 4.1.5.1.3: Examples of Active and Inert Maps of Pointed Sets
    • Proposition 4.1.5.1.4: Properties of Active and Inert Maps of Pointed Sets
• Section 4.2: Limits of Pointed Sets
  • Subsection 4.2.1: The Terminal Pointed Set
    • Definition 4.2.1.1.1: The Terminal Pointed Set
  • Subsection 4.2.2: Products of Families of Pointed Sets
    • Definition 4.2.2.1.1: The Product of a Family of Pointed Sets
    • Proposition 4.2.2.1.2: Properties of Products of Families of Pointed Sets
  • Subsection 4.2.3: Products
    • Definition 4.2.3.1.1: Products of Pointed Sets
    • Proposition 4.2.3.1.2: Properties of Products of Pointed Sets
  • Subsection 4.2.4: Pullbacks
    • Definition 4.2.4.1.1: Pullbacks of Pointed Sets
    • Proposition 4.2.4.1.2: Properties of Pullbacks of Pointed Sets
  • Subsection 4.2.5: Equalisers
    • Definition 4.2.5.1.1: Equalisers of Pointed Sets
    • Proposition 4.2.5.1.2: Properties of Equalisers of Pointed Sets
• Section 4.3: Colimits of Pointed Sets
  • Subsection 4.3.1: The Initial Pointed Set
    • Definition 4.3.1.1.1: The Initial Pointed Set
  • Subsection 4.3.2: Coproducts of Families of Pointed Sets
    • Definition 4.3.2.1.1: Coproducts of Families of Pointed Sets
    • Proposition 4.3.2.1.2: Properties of Coproducts of Families of Pointed Sets
  • Subsection 4.3.3: Coproducts
    • Definition 4.3.3.1.1: Coproducts of Pointed Sets
    • Proposition 4.3.3.1.2: Properties of Wedge Sums of Pointed Sets
  • Subsection 4.3.4: Pushouts
    • Definition 4.3.4.1.1: Pushouts of Pointed Sets
    • Proposition 4.3.4.1.2: Properties of Pushouts of Pointed Sets
  • Subsection 4.3.5: Coequalisers
    • Definition 4.3.5.1.1: Coequalisers of Pointed Sets
    • Proposition 4.3.5.1.2: Properties of Coequalisers of Pointed Sets
• Section 4.4: Constructions With Pointed Sets
  • Subsection 4.4.1: Free Pointed Sets
    • Definition 4.4.1.1.1: Free Pointed Sets
    • Proposition 4.4.1.1.2: Properties of Free Pointed Sets
  • Subsection 4.4.2: Deleting Basepoints
    • Definition 4.4.2.1.1: Sets With Deleted Basepoints
    • Proposition 4.4.2.1.2: Properties of Sets With Deleted Basepoints