The Clowder Project

7.5 The Smash Product of Pointed Sets

• Subsection 7.5.1: Foundations
  • Definition 7.5.1.1.1: Smash Products of Pointed Sets
  • Remark 7.5.1.1.2: Unwinding Definition 7.5.1.1.1: The Universal Property I
  • Remark 7.5.1.1.3: Unwinding Definition 7.5.1.1.1: The Universal Property II
  • Construction 7.5.1.1.4: Smash Products of Pointed Sets
  • Remark 7.5.1.1.5: On the Construction of the Smash Product of Pointed Sets
  • Construction 7.5.1.1.6: A Second Construction of the Smash Product of Pointed Sets
  • Notation 7.5.1.1.7: Elements of Smash Products of Pointed Sets
  • Remark 7.5.1.1.8: Basepoints of Smash Products of Pointed Sets
  • Example 7.5.1.1.9: Examples of Smash Products of Pointed Sets
  • Proposition 7.5.1.1.10: Properties of Smash Products of Pointed Sets
• Subsection 7.5.2: The Internal Hom of Pointed Sets
  • Definition 7.5.2.1.1: The Internal Hom of Pointed Sets
  • Proposition 7.5.2.1.2: Properties of the Internal Hom of Pointed Sets
• Subsection 7.5.3: The Monoidal Unit
  • Definition 7.5.3.1.1: The Monoidal Unit of $\wedge $
• Subsection 7.5.4: The Associator
  • Definition 7.5.4.1.1: The Associator of $\wedge $
• Subsection 7.5.5: The Left Unitor
  • Definition 7.5.5.1.1: The Left Unitor of $\wedge $
• Subsection 7.5.6: The Right Unitor
  • Definition 7.5.6.1.1: The Right Unitor of $\wedge $
• Subsection 7.5.7: The Symmetry
  • Definition 7.5.7.1.1: The Symmetry of $\wedge $
• Subsection 7.5.8: The Diagonal
  • Definition 7.5.8.1.1: The Diagonal of $\wedge $
  • Proposition 7.5.8.1.2: Properties of the Diagonal of $\wedge $
• Subsection 7.5.9: The Monoidal Structure on Pointed Sets Associated to $\wedge $
  • Proposition 7.5.9.1.1: The Monoidal Structure on Pointed Sets Associated to $\wedge $
• Subsection 7.5.10: The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
  • Theorem 7.5.10.1.1: The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
  • Corollary 7.5.10.1.2: A Second Universal Property for $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
  • Corollary 7.5.10.1.3: A Third Universal Property of the Smash Product of Pointed Sets
• Subsection 7.5.11: Monoids With Respect to the Smash Product of Pointed Sets
  • Proposition 7.5.11.1.1: Monoids With Respect to $\wedge $
• Subsection 7.5.12: Comonoids With Respect to the Smash Product of Pointed Sets
  • Proposition 7.5.12.1.1: Comonoids With Respect to $\wedge $