5 Tensor Products of Pointed Sets
In this chapter we introduce, construct, and study tensor products of pointed sets. The most well-known among these is the smash product of pointed sets
\[ \wedge \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*}, \]
introduced in Section 5.5.1, defined via a universal property as inducing a bijection between the following data:
- Pointed maps $f\colon X\wedge Y\to Z$.
- Maps of sets $f\colon X\times Y\to Z$ satisfying
\begin{align*} f\webleft (x_{0},y\webright ) & = z_{0},\\ f\webleft (x,y_{0}\webright ) & = z_{0} \end{align*}
for each $x\in X$ and each $y\in Y$.
As it turns out, however, dropping either of the
bilinearity conditions
\begin{align*} f\webleft (x_{0},y\webright ) & = z_{0},\\ f\webleft (x,y_{0}\webright ) & = z_{0} \end{align*}
while retaining the other leads to two other tensor products of pointed sets,
\begin{align*} \lhd & \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ \rhd & \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}, \end{align*}
called the left and right tensor products of pointed sets. In contrast to $\wedge $, which turns out to endow $\mathsf{Sets}_{*}$ with a monoidal category structure (Proposition 5.5.9.1.1), these do not admit invertible associators and unitors, but do endow $\mathsf{Sets}_{*}$ with the structure of a skew monoidal category, however (Proposition 5.3.8.1.1 and Proposition 5.4.8.1.1).
Finally, in addition to the tensor products $\lhd $, $\rhd $, and $\wedge $, we also have a “tensor product” of the form
\[ \odot \colon \mathsf{Sets}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*}, \]
called the tensor of sets with pointed sets. All in all, these tensor products assemble into a family of functors of the form
\begin{align*} \otimes _{k,\ell } & \colon \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright )\times \mathsf{Mon}_{\mathbb {E}_{\ell }}\webleft (\mathsf{Sets}\webright ) \to \mathsf{Mon}_{\mathbb {E}_{k+\ell }}\webleft (\mathsf{Sets}\webright ),\\ \lhd _{i,k} & \colon \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright )\times \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright ) \to \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright ),\\ \rhd _{i,k} & \colon \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright )\times \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright ) \to \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright ),\end{align*}
where $k,\ell ,i\in \mathbb {N}$ with $i\leq k-1$. Together with the Cartesian product $\times $ of $\mathsf{Sets}$, the tensor products studied in this chapter form the cases:
- $\webleft (k,\ell \webright )=\webleft (-1,-1\webright )$ for the Cartesian product of $\mathsf{Sets}$;
- $\webleft (k,\ell \webright )=\webleft (0,-1\webright )$ and $\webleft (-1,0\webright )$ for the tensor of sets with pointed sets of Definition 5.2.1.1.1;
- $\webleft (i,k\webright )=\webleft (-1,0\webright )$ for the left and right tensor products of pointed sets of Section 5.3 and Section 5.4;
- $\webleft (k,\ell \webright )=\webleft (-1,-1\webright )$ for the smash product of pointed sets of Section 5.5.
In this chapter, we will carefully define and study bilinearity for pointed sets, as well as all the tensor products described above. Then, in
, we will extend these to tensor products involving also monoids and commutative monoids, which will end up covering all cases up to $k,\ell \leq 2$, and hence
all cases since $\mathbb {E}_{k}$-monoids on $\mathsf{Sets}$ are the same as $\mathbb {E}_{2}$-monoids on $\mathsf{Sets}$ when $k\geq 2$.
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Section 5.1: Bilinear Morphisms of Pointed Sets
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Subsection 5.1.1: Left Bilinear Morphisms of Pointed Sets
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Definition 5.1.1.1.1: Left Bilinear Morphisms of pointed sets
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Definition 5.1.1.1.2: The Set of Left Bilinear Morphisms of Pointed Sets
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Subsection 5.1.2: Right Bilinear Morphisms of Pointed Sets
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Definition 5.1.2.1.1: Right Bilinear Morphisms of pointed sets
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Definition 5.1.2.1.2: The Set of Right Bilinear Morphisms of Pointed Sets
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Subsection 5.1.3: Bilinear Morphisms of Pointed Sets
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Section 5.2: Tensors and Cotensors of Pointed Sets by Sets
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Subsection 5.2.1: Tensors of Pointed Sets by Sets
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Definition 5.2.1.1.1: Tensors of Pointed Sets by Sets
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Remark 5.2.1.1.2: Unwinding Definition 5.2.1.1.1
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Construction 5.2.1.1.3: Construction of Tensors of Pointed Sets by Sets
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Notation 5.2.1.1.4: Elements of Tensors of Pointed Sets by Sets
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Remark 5.2.1.1.5: Basepoints of Tensors of Pointed Sets by Sets
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Proposition 5.2.1.1.6: Properties of Tensors of Pointed Sets by Sets
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Subsection 5.2.2: Cotensors of Pointed Sets by Sets
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Section 5.3: The Left Tensor Product of Pointed Sets
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Subsection 5.3.1: Foundations
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Subsection 5.3.2: The Left Internal Hom of Pointed Sets
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Subsection 5.3.3: The Left Skew Unit
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Definition 5.3.3.1.1: The Left Skew Unit of $\lhd $
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Subsection 5.3.4: The Left Skew Associator
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Subsection 5.3.5: The Left Skew Left Unitor
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Subsection 5.3.6: The Left Skew Right Unitor
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Subsection 5.3.7: The Diagonal
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Definition 5.3.7.1.1: The Diagonal of $\lhd $
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Subsection 5.3.8: The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $
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Proposition 5.3.8.1.1: The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $
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Subsection 5.3.9: Monoids With Respect to the Left Tensor Product of Pointed Sets
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Proposition 5.3.9.1.1: Monoids With Respect to $\lhd $
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Section 5.4: The Right Tensor Product of Pointed Sets
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Subsection 5.4.1: Foundations
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Subsection 5.4.2: The Right Internal Hom of Pointed Sets
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Subsection 5.4.3: The Right Skew Unit
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Definition 5.4.3.1.1: The Right Skew Unit of $\rhd $
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Subsection 5.4.4: The Right Skew Associator
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Subsection 5.4.5: The Right Skew Left Unitor
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Subsection 5.4.6: The Right Skew Right Unitor
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Subsection 5.4.7: The Diagonal
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Definition 5.4.7.1.1: The Diagonal of $\rhd $
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Subsection 5.4.8: The Right Skew Monoidal Structure on Pointed Sets Associated to $\rhd $
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Proposition 5.4.8.1.1: The Right Skew Monoidal Structure on Pointed Sets Associated to $\rhd $
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Subsection 5.4.9: Monoids With Respect to the Right Tensor Product of Pointed Sets
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Proposition 5.4.9.1.1: Monoids With Respect to $\rhd $
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Section 5.5: The Smash Product of Pointed Sets
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Subsection 5.5.1: Foundations
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Definition 5.5.1.1.1: Smash Products of Pointed Sets
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Remark 5.5.1.1.2: Unwinding Definition 5.5.1.1.1: The Universal Property I
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Remark 5.5.1.1.3: Unwinding Definition 5.5.1.1.1: The Universal Property II
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Construction 5.5.1.1.4: Smash Products of Pointed Sets
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Remark 5.5.1.1.5: On the Construction of the Smash Product of Pointed Sets
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Construction 5.5.1.1.6: A Second Construction of the Smash Product of Pointed Sets
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Notation 5.5.1.1.7: Elements of Smash Products of Pointed Sets
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Remark 5.5.1.1.8: Basepoints of Smash Products of Pointed Sets
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Example 5.5.1.1.9: Examples of Smash Products of Pointed Sets
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Proposition 5.5.1.1.10: Properties of Smash Products of Pointed Sets
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Subsection 5.5.2: The Internal Hom of Pointed Sets
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Definition 5.5.2.1.1: The Internal Hom of Pointed Sets
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Proposition 5.5.2.1.2: Properties of the Internal Hom of Pointed Sets
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Subsection 5.5.3: The Monoidal Unit
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Definition 5.5.3.1.1: The Monoidal Unit of $\wedge $
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Subsection 5.5.4: The Associator
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Definition 5.5.4.1.1: The Associator of $\wedge $
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Subsection 5.5.5: The Left Unitor
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Definition 5.5.5.1.1: The Left Unitor of $\wedge $
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Subsection 5.5.6: The Right Unitor
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Definition 5.5.6.1.1: The Right Unitor of $\wedge $
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Subsection 5.5.7: The Symmetry
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Definition 5.5.7.1.1: The Symmetry of $\wedge $
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Subsection 5.5.8: The Diagonal
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Definition 5.5.8.1.1: The Diagonal of $\wedge $
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Proposition 5.5.8.1.2: Properties of the Diagonal of $\wedge $
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Subsection 5.5.9: The Monoidal Structure on Pointed Sets Associated to $\wedge $
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Proposition 5.5.9.1.1: The Monoidal Structure on Pointed Sets Associated to $\wedge $
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Subsection 5.5.10: The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
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Theorem 5.5.10.1.1: The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
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Corollary 5.5.10.1.2: A Second Universal Property for $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$
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Corollary 5.5.10.1.3: A Third Universal Property of the Smash Product of Pointed Sets
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Subsection 5.5.11: Monoids With Respect to the Smash Product of Pointed Sets
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Proposition 5.5.11.1.1: Monoids With Respect to $\wedge $
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Subsection 5.5.12: Comonoids With Respect to the Smash Product of Pointed Sets
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Proposition 5.5.12.1.1: Comonoids With Respect to $\wedge $
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Section 5.6: Miscellany
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Subsection 5.6.1: The Smash Product of a Family of Pointed Sets
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Definition 5.6.1.1.1: The Smash Product of a Family of Pointed Sets