5 Monoidal Structures on the Category of Sets
This chapter contains some material on monoidal structures on $\mathsf{Sets}$.
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Section 5.1: The Monoidal Category of Sets and Products
- Subsection 5.1.1: Products of Sets
- Subsection 5.1.2: The Internal Hom of Sets
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Subsection 5.1.3: The Monoidal Unit
- Definition 5.1.3.1.1: The Monoidal Unit of $\times $
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Subsection 5.1.4: The Associator
- Definition 5.1.4.1.1: The Associator of $\times $
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Subsection 5.1.5: The Left Unitor
- Definition 5.1.5.1.1: The Left Unitor of $\times $
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Subsection 5.1.6: The Right Unitor
- Definition 5.1.6.1.1: The Right Unitor of $\times $
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Subsection 5.1.7: The Symmetry
- Definition 5.1.7.1.1: The Symmetry of $\times $
- Subsection 5.1.8: The Diagonal
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Subsection 5.1.9: The Monoidal Category of Sets and Products
- Proposition 5.1.9.1.1: The Monoidal Structure on Sets Associated to the Product
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Subsection 5.1.10: The Universal Property of $\webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright )$
- Theorem 5.1.10.1.1: The Universal Property of $\webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright )$
- Corollary 5.1.10.1.2: A Second Universal Property for $\webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright )$
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Section 5.2: The Monoidal Category of Sets and Coproducts
- Subsection 5.2.1: Coproducts of Sets
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Subsection 5.2.2: The Monoidal Unit
- Definition 5.2.2.1.1: The Monoidal Unit of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.2.3: The Associator
- Definition 5.2.3.1.1: The Associator of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.2.4: The Left Unitor
- Definition 5.2.4.1.1: The Left Unitor of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.2.5: The Right Unitor
- Definition 5.2.5.1.1: The Right Unitor of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.2.6: The Symmetry
- Definition 5.2.6.1.1: The Symmetry of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.2.7: The Monoidal Category of Sets and Coproducts
- Proposition 5.2.7.1.1: The Monoidal Structure on Sets Associated to $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Section 5.3: The Bimonoidal Category of Sets, Products, and Coproducts
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Subsection 5.3.1: The Left Distributor
- Definition 5.3.1.1.1: The Left Distributor of $\times $ over $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.3.2: The Right Distributor
- Definition 5.3.2.1.1: The Right Distributor of $\times $ over $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.3.3: The Left Annihilator
- Definition 5.3.3.1.1: The Left Annihilator of $\times $
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Subsection 5.3.4: The Right Annihilator
- Definition 5.3.4.1.1: The Right Annihilator of $\times $
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Subsection 5.3.5: The Bimonoidal Category of Sets, Products, and Coproducts
- Proposition 5.3.5.1.1: The Bimonoidal Structure on Sets Associated to $\times $ and $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
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Subsection 5.3.1: The Left Distributor
