The Clowder Project

4.3 The Left Tensor Product of Pointed Sets

• Subsection 4.3.1: Foundations
  • Definition 4.3.1.1.1: The Left Tensor Product of Pointed Sets
  • Remark 4.3.1.1.2: Unwinding Definition 4.3.1.1.1: Universal Property I
  • Remark 4.3.1.1.3: Unwinding Definition 4.3.1.1.1: Universal Property II
  • Construction 4.3.1.1.4: The Left Tensor Product of Pointed Sets
  • Notation 4.3.1.1.5: Elements of Left Tensor Products of Pointed Sets
  • Remark 4.3.1.1.6: Basepoints of Left Tensor Products of Pointed Sets
  • Proposition 4.3.1.1.7: Properties of Left Tensor Products of Pointed Sets
  • Remark 4.3.1.1.8: On the Failure of $X\lhd -$ To Be a Left Adjoint
• Subsection 4.3.2: The Left Internal Hom of Pointed Sets
  • Definition 4.3.2.1.1: The Left Internal Hom of Pointed Sets
  • Remark 4.3.2.1.2: Unwinding Definition 4.3.2.1.1, I: Universal Property
  • Remark 4.3.2.1.3: Unwinding Definition 4.3.2.1.1, II: Explicit Description
  • Proposition 4.3.2.1.4: Properties of Left Internal Homs of Pointed Sets
• Subsection 4.3.3: The Left Skew Unit
  • Definition 4.3.3.1.1: The Left Skew Unit of $\lhd $
• Subsection 4.3.4: The Left Skew Associator
  • Definition 4.3.4.1.1: The Left Skew Associator of $\lhd $
  • Remark 4.3.4.1.2: Unwinding Definition 4.3.4.1.1
  • Remark 4.3.4.1.3: Non-Invertibility of the Skew Associator of $\lhd $
• Subsection 4.3.5: The Left Skew Left Unitor
  • Definition 4.3.5.1.1: The Left Skew Left Unitor of $\lhd $
  • Remark 4.3.5.1.2: Unwinding Definition 4.3.5.1.1
  • Remark 4.3.5.1.3: Non-Invertibility of the Skew Left Unitor of $\lhd $
• Subsection 4.3.6: The Left Skew Right Unitor
  • Definition 4.3.6.1.1: The Left Skew Right Unitor of $\lhd $
  • Remark 4.3.6.1.2: Unwinding Definition 4.3.6.1.1
  • Remark 4.3.6.1.3: Non-Invertibility of the Skew Right Unitor of $\lhd $
• Subsection 4.3.7: The Diagonal
  • Definition 4.3.7.1.1: The Diagonal of $\lhd $
• Subsection 4.3.8: The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $
  • Proposition 4.3.8.1.1: The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $
• Subsection 4.3.9: Monoids With Respect to the Left Tensor Product of Pointed Sets
  • Proposition 4.3.9.1.1: Monoids With Respect to $\lhd $