7.5.10 The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$

    Theorem 7.5.10.1.1 ▶ The Universal Property of $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$

    The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ of Proposition 7.5.9.1.1 is uniquely determined by the following requirements:

    1. 1.

      Existence of an Internal Hom. The tensor product

      \[ \otimes _{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]

      of $\mathsf{Sets}_{*}$ admits an internal Hom $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$.

  • 2.

    The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}\cong S^{0}$.

    • More precisely, the full subcategory of the category $\mathcal{M}^{\mathrm{cld}}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of Unresolved reference spanned by the closed symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ satisfying Item 1 and Item 2 is contractible (i.e. equivalent to the punctual category).

    Proof of Theorem 7.5.10.1.1.

    Unwinding the Statement
    Let $\webleft (\mathsf{Sets}_{*},\otimes _{\mathsf{Sets}_{*}},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}},\mathbb {1}_{\mathsf{Sets}_{*}},\lambda ',\rho ',\sigma '\webright )$ be a closed symmetric monoidal category satisfying Item 1 and Item 2. We need to show that the identity functor

    \[ \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

    admits a unique closed symmetric monoidal functor structure

    \[ \begin{array}{cccc} \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}} \colon \mkern -10mu & X\otimes _{\mathsf{Sets}_{*}}Y \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}X\wedge Y,\\ \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\operatorname {\mathrm{Hom}}}_{\mathsf{Sets}_{*}}} \colon \mkern -10mu & \webleft [X,Y\webright ]_{\mathsf{Sets}_{*}} \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}\mathsf{Sets}_{*}\webleft (X,Y\webright ),\\ \phantom{\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}}\mathllap {\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}} \colon \mkern -10mu & \mathbb {1}_{\mathsf{Sets}_{*}} \mkern -10mu& {}\mathbin {\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}& \mkern -10mu{}S^{0}, \end{array} \]

    making it into a symmetric monoidal strongly closed isomorphism of categories from $\left(\phantom{\mathrlap {\lambda '}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda '$, $\rho '$, $\left.\sigma '\right)$ to the closed symmetric monoidal category $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\times $, $\mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ of Proposition 7.5.9.1.1.

    Constructing an Isomorphism $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\cong \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$
    By Unresolved reference, we have a natural isomorphism

    \[ \mathsf{Sets}_{*}\webleft (S^{0},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\webright )\cong \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright ). \]

    By Chapter 6: Pointed Sets, Item 4 of Proposition 6.1.4.1.1, we also have a natural isomorphism

    \[ \mathsf{Sets}_{*}\webleft (S^{0},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}\webright )\cong \webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}. \]

    Composing both natural isomorphisms, we obtain a natural isomorphism

    \[ \mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )\cong \webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}. \]

    Given $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, we will write

    \[ \operatorname {\mathrm{id}}^{\operatorname {\mathrm{Hom}}}_{X,Y}\colon \mathsf{Sets}_{*}\webleft (X,Y\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft [X,Y\webright ]_{\mathsf{Sets}_{*}} \]

    for the component of this isomorphism at $\webleft (X,Y\webright )$.

    Constructing an Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    Since $\otimes _{\mathsf{Sets}_{*}}$ is adjoint in each variable to $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}_{*}}$ by assumption and $\wedge $ is adjoint in each variable to $\mathsf{Sets}_{*}\webleft (-_{1},-_{2}\webright )$ by Chapter 4: Constructions With Sets, Item 2 of Proposition 4.3.5.1.2, uniqueness of adjoints (Unresolved reference) gives us natural isomorphisms

    \begin{align*} X\otimes _{\mathsf{Sets}_{*}}- & \cong X\wedge -,\\ -\otimes _{\mathsf{Sets}_{*}}Y & \cong Y\wedge -. \end{align*}

    By Unresolved reference, we then have $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$. We will write

    \[ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\colon X\otimes _{\mathsf{Sets}_{*}}Y\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\wedge Y \]

    for the component of this isomorphism at $\webleft (X,Y\webright )$.

    Alternative Construction of an Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    Alternatively, we may construct a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$ as follows:

    1. 1.

      Let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.

    2. 2.

      Since $\otimes _{\mathsf{Sets}_{*}}$ is part of a closed monoidal structure, it preserves colimits in each variable by Unresolved reference.

    3. 3.

      Since $X\cong \bigvee _{x\in X^{-}}S^{0}$ and $\otimes _{\mathsf{Sets}_{*}}$ preserves colimits in each variable, we have

      \begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\otimes _{\mathsf{Sets}_{*}}Y\\ & \cong \bigvee _{x\in X^{-}}\webleft (S^{0}\otimes _{\mathsf{Sets}_{*}}Y\webright )\\ & \cong \bigvee _{x\in X^{-}}Y\\ & \cong \bigvee _{x\in X^{-}}S^{0}\wedge Y\\ & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\wedge Y\\ & \cong X\wedge Y, \end{align*}

      naturally in $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, where we have used that $S^{0}$ is the monoidal unit for $\otimes _{\mathsf{Sets}_{*}}$. Thus $X\otimes _{\mathsf{Sets}_{*}}-\cong X\wedge -$ for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.

    4. 4.

      Similarly, $-\otimes _{\mathsf{Sets}_{*}}Y\cong -\wedge Y$ for each $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.

    5. 5.

      By Unresolved reference, we then have $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$.

    Below, we’ll show that if a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$ exists, then it must be unique. This will show that the isomorphism constructed above is equal to the isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\colon X\otimes _{\mathsf{Sets}_{*}}Y\to X\wedge Y$ from before.

    Constructing an Isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}_{*}}\to S^{0}$
    We define an isomorphism $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}_{*}}\to S^{0}$ as the composition

    \[ \mathbb {1}_{\mathsf{Sets}_{*}}\overset {\rho ^{\mathsf{Sets}_{*},-1}_{\mathbb {1}_{\mathsf{Sets}_{*}}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}\mathbb {1}_{\mathsf{Sets}_{*}}\wedge S^{0}\overset {\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|\mathbb {1}_{\mathsf{Sets}_{*}}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}\mathbb {1}_{\mathsf{Sets}_{*}}\otimes _{\mathsf{Sets}_{*}} S^{0}\overset {\lambda '_{ S^{0}}}{\underset {\scriptstyle \mathord {\sim }}{\dashrightarrow }}S^{0} \]

    in $\mathsf{Sets}_{*}$.

    Monoidal Left Unity of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    We have to show that the diagram
    commutes. To this end, we will first show that the diagram
    corresponding to the case $X=S^{0}$, commutes. Indeed, consider the diagram
    whose boundary diagram corresponds to the diagram $\webleft (\dagger \webright )$ above. In this diagram:

    • •

      Subdiagrams $\webleft (1\webright )$, $\webleft (2\webright )$, and $\webleft (3\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (4\webright )$ commutes by Unresolved reference.

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*},-1}$.

    • •

      Subdiagram $\webleft (6\webright )$ commutes trivially.

    • •

      Subdiagram $\webleft (7\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*}}$, where the equality $\rho ^{\mathsf{Sets}_{*}}_{S^{0}}=\lambda ^{\mathsf{Sets}_{*}}_{S^{0}}$ comes from Unresolved reference.

    Since all subdiagrams commute, so does the boundary diagram, i.e. the diagram $\webleft (\dagger \webright )$ above. As a result, the diagram

    also commutes. Now, let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, let $x\in X$, and consider the diagram
    Since:

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\lambda ^{\prime ,-1}$.

    • •

      Subdiagram $\webleft (\ddagger \webright )$ commutes, as proved above.

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathsf{Sets}_{*},-1}$.

    it follows that the diagram

    Here’s a step-by-step showcase of this argument:
    We then have

    \begin{align*} \lambda ^{\prime ,-1}_{X}\webleft (x\webright ) & = \webleft [\lambda ^{\prime ,-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webright ]\webleft (x\webright ) \end{align*}

    for each $x\in X$, and thus we have

    \[ \lambda ^{\prime ,-1}_{X}=\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}. \]

    Taking inverses then gives

    \[ \lambda ^{\prime }_{X}=\lambda ^{\mathsf{Sets}_{*}}_{X}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|S^{0},X}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\wedge \operatorname {\mathrm{id}}_{X}\webright ), \]

    showing that the diagram

    indeed commutes.

    Braidedness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    We have to show that the diagram
    commutes. To this end, we will first show that the diagram
    commutes. To that end, we will first show that the diagram
    commutes, and, to this end, we will first show that the diagram
    commutes. Indeed, consider the diagram
    whose boundary diagram corresponds to diagram $\webleft (\S \webright )$ above. Since:

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;

    • •

      Subdiagrams $\webleft (2\webright )$ and $\webleft (3\webright )$ commute by the functoriality of $\otimes $;

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the left monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes },\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\webright )$, which we proved above;

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\lambda '$;

    • •

      Subdiagram $\webleft (6\webright )$ commutes by the naturality of $\rho '$, where the equality $\rho '_{\mathbb {1}_{\mathsf{Sets}_{*}}}=\lambda '_{\mathbb {1}_{\mathsf{Sets}_{*}}}$ comes from Unresolved reference;

    it follows that the boundary diagram, i.e. diagram $\webleft (\S \webright )$, also commutes. Next, consider the diagram

    whose boundary diagram corresponds to the diagram $\webleft (\ddagger \webright )$ above. Since:

    • •

      Subdiagrams $\webleft (1\webright )$ and $\webleft (6\webright )$ commute by Unresolved reference;

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;

    • •

      Subdiagram $\webleft (\S \webright )$ commutes, as was shown above;

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathsf{Sets}_{*}}$;

    • •

      Subdiagram $\webleft (4\webright )$ commutes trivially;

    • •

      Subdiagram $\webleft (5\webright )$ commutes by Chapter 12: Constructions With Monoidal Categories, Item 2c of Item 2 of Proposition 12.1.1.1.4, whose proof uses only the left monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes },\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}\webright )$, which has been proven above;

    it follows that the boundary diagram, i.e. diagram $\webleft (\ddagger \webright )$, also commutes. Next, consider the diagram

    whose boundary diagram corresponds to the diagram $\webleft (\dagger \webright )$. Since:

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\sigma '$ and the fact that $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}$ is invertible;

    • •

      Subdiagram $\webleft (\ddagger \webright )$ commutes as proved above;

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\sigma ^{\mathsf{Sets}_{*}}$ and the fact that $\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}}$ is invertible;

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;

    it follows that the boundary diagram, i.e. diagram $\webleft (\dagger \webright )$ also commutes. Taking inverses for the diagram $\webleft (\dagger \webright )$, we see that the diagram

    commutes as well. Now, let $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, let $x\in X$, let $y\in Y$, and consider the diagram
    which we partition into subdiagrams as follows:
    Since:

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\sigma ^{\mathsf{Sets}_{*},-1}$.

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}$.

    • •

      Subdiagram $\webleft (¶\webright )$ commutes, as proved above.

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\sigma ^{\prime ,-1}$.

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}$.

    it follows that the diagram

    commutes. We then have

    \begin{align*} \webleft [\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}\webright ]\webleft (y,x\webright ) & = \webleft [\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}\circ \webleft (\webleft [y\webright ]\wedge \webleft [x\webright ]\webright )\webright ]\webleft (1,1\webright )\\ & = \webleft [\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}\circ \webleft (\webleft [y\webright ]\wedge \webleft [x\webright ]\webright )\webright ]\webleft (1,1\webright )\\ & = \webleft [\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}\webright ]\webleft (y,x\webright ) \end{align*}

    for each $\webleft (y,x\webright )\in Y\wedge X$, and thus we have

    \[ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\circ \sigma ^{\mathsf{Sets}_{*},-1}_{X,Y}=\sigma ^{\prime ,-1}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,X}. \]

    Taking inverses then gives

    \[ \sigma ^{\mathsf{Sets}_{*}}_{X,Y}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}=\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|Y,X}\circ \sigma ^{\prime }_{X,Y}, \]

    showing that the diagram

    indeed commutes.

    Monoidal Right Unity of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    We have to show that the diagram
    commutes. To this end, we will first show that the diagram
    corresponding to the case $X=S^{0}$, commutes. First, notice that we may write

    \[ \sigma '_{S^{0},\mathbb {1}_{\mathsf{Sets}_{*}}}\colon S^{0}\otimes _{\mathsf{Sets}_{*}}\mathbb {1}_{\mathsf{Sets}_{*}}\to \mathbb {1}_{\mathsf{Sets}_{*}}\otimes _{\mathsf{Sets}_{*}}S^{0} \]

    as the composition

    Indeed, we may write this composition as part of the diagram
    which commutes since:

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the braidedness of $\operatorname {\mathrm{id}}^{\otimes }$, as proved above.

    • •

      Subdiagram $\webleft (2\webright )$ commutes by Unresolved reference.

    Next, consider the diagram

    whose boundary diagram corresponds to the diagram $\webleft (\dagger \webright )$ above, since the composition in red is equal to $\sigma '_{S^{0},\mathbb {1}_{\mathsf{Sets}_{*}}}$ as proved above, and then the composition in red composed with $\lambda '_{S^{0}}$ is equal to $\rho '_{S^{0}}$ by Unresolved reference. In this diagram:

    • •

      Subdiagrams $\webleft (1\webright )$, $\webleft (2\webright )$, and $\webleft (3\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagrams $\webleft (4\webright )$, $\webleft (5\webright )$, and $\webleft (6\webright )$ commute by the naturality of $\lambda ^{\mathsf{Sets}_{*}}$, where the equality $\lambda ^{\mathsf{Sets}_{*}}_{S^{0}}=\rho ^{\mathsf{Sets}_{*}}_{S^{0}}$ comes from Unresolved reference.

    Since all subdiagrams commute, so does the boundary diagram, i.e. the diagram $\webleft (\dagger \webright )$ above. As a result, the diagram

    also commutes. Now, let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, let $x\in X$, and consider the diagram
    Since:

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\rho ^{\prime ,-1}$.

    • •

      Subdiagram $\webleft (\dagger \webright )$ commutes, as proved above.

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*},-1}$.

    it follows that the diagram

    Here’s a step-by-step showcase of this argument:
    We then have

    \begin{align*} \rho ^{\prime ,-1}_{X}\webleft (a\webright ) & = \webleft [\rho ^{\prime ,-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}\circ \webleft [x\webright ]\webright ]\webleft (1\webright )\\ & = \webleft [\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}\webright ]\webleft (a\webright ) \end{align*}

    for each $a\in X$, and thus we have

    \[ \rho ^{\prime ,-1}_{X}=\webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|S^{0},X}\circ \rho ^{\mathsf{Sets}_{*},-1}_{X}. \]

    Taking inverses then gives

    \[ \rho ^{\prime }_{X}=\rho ^{\mathsf{Sets}_{*}}_{X}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|S^{0},X}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\webright ), \]

    showing that the diagram

    indeed commutes.

    Monoidality of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    We have to show that the diagram
    commutes. To this end, we will first prove that the diagram
    commutes, and, to that end, we will first show that the diagram
    commutes. Indeed, consider the diagram
    whose boundary diagram corresponds to diagram $\webleft (\ddagger \webright )$ above. Since:

    • •

      Subdiagrams $\webleft (1\webright )$, $\webleft (4\webright )$, $\webleft (5\webright )$, $\webleft (8\webright )$, and $\webleft (11\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}}$;

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the right monoidal unity of $\webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}_{*}}\webright )$;

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the triangle identity for $\webleft (\alpha ',\lambda ',\rho '\webright )$;

    • •

      Subdiagram $\webleft (6\webright )$ commutes by Unresolved reference;

    • •

      Subdiagram $\webleft (7\webright )$ commutes by the naturality of $\rho ^{\mathsf{Sets}_{*}}$;

    • •

      Subdiagram $\webleft (9\webright )$ commutes by Unresolved reference;

    • •

      Subdiagram $\webleft (10\webright )$ commutes by Unresolved reference;

    it follows that the boundary diagram, i.e. diagram $\webleft (\ddagger \webright )$, also commutes. Consider now the diagram

    whose boundary corresponds to diagram $\webleft (\dagger \webright )$ above. Since:

    • •

      Subdiagrams $\webleft (1\webright )$, $\webleft (3\webright )$, $\webleft (4\webright )$, and $\webleft (6\webright )$ commute by the naturality of $\operatorname {\mathrm{id}}^{\otimes }_{Sets_{*}}$;

    • •

      Subdiagram $\webleft (\ddagger \webright )$ commutes, as proved above;

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\alpha '$;

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\alpha ^{\mathsf{Sets}_{*}}$;

    it follows that the boundary diagram, i.e. diagram $\webleft (\dagger \webright )$, also commutes. Taking inverses on the diagram $\webleft (\dagger \webright )$, we see that the diagram

    commutes as well. Now, let $X,Y,Z\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, let $x\in X$, let $y\in Y$, let $z\in Z$, and consider the diagram
    which we partition into subdiagrams as follows:
    Since:

    • •

      Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\alpha ^{\mathsf{Sets}_{*},-1}$.

    • •

      Subdiagram $\webleft (2\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (\dagger \webright )$ commutes, as proved above.

    • •

      Subdiagram $\webleft (4\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (5\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}}$.

    • •

      Subdiagram $\webleft (6\webright )$ commutes by the naturality of $\alpha ^{\prime ,-1}$.

    it follows that the diagram

    also commutes. We then have
    \begin{align*} \left[\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\right.\\ \left.{}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}\right]\webleft (x,\webleft (y,z\webright )\webright ) & = \left[\webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\right.\\ & \phantom{={}} \mkern 4mu\left.{}{}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}\circ \webleft (\webleft [x\webright ]\wedge \webleft (\webleft [y\webright ]\wedge \webleft [z\webright ]\webright )\webright )\right]\webleft (1,\webleft (1,1\webright )\webright )\\ & = \left[\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\right.\\ & \phantom{={}} \mkern 4mu\left.{}\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}\circ \webleft (\webleft [x\webright ]\wedge \webleft (\webleft [y\webright ]\wedge \webleft [z\webright ]\webright )\webright )\right]\webleft (1,\webleft (1,1\webright )\webright )\\ & = \webleft [\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}\webright ]\webleft (x,\webleft (y,z\webright )\webright ) \end{align*}
    for each $\webleft (x,\webleft (y,z\webright )\webright )\in X\wedge \webleft (Y\wedge Z\webright )$, and thus we have
    \[ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X\wedge Y,Z}\circ \alpha ^{\mathsf{Sets}_{*},-1}_{X,Y,Z}=\alpha ^{\prime ,-1}_{X,Y,Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathsf{Sets}_{*}|X,Y\wedge Z}. \]
    Taking inverses then gives
    \[ \alpha ^{\mathsf{Sets}_{*}}_{X,Y,Z}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X\wedge Y,Z}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y}\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Z}\webright )=\operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|X,Y\wedge Z}\circ \webleft (\operatorname {\mathrm{id}}_{X}\wedge \operatorname {\mathrm{id}}^{\otimes }_{\mathsf{Sets}_{*}|Y,Z}\webright )\circ \alpha ^{\prime }_{X,Y,Z}, \]
    showing that the diagram
    indeed commutes.

    Uniqueness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$
    Let $\phi ,\psi \colon -_{1}\otimes _{\mathsf{Sets}_{*}}-_{2}\Rightarrow -_{1}\wedge -_{2}$ be natural isomorphisms. Since these isomorphisms are compatible with the unitors of $\mathsf{Sets}_{*}$ with respect to $\wedge $ and $\otimes $ (as shown above), we have

    \begin{align*} \lambda '_{Y} & = \lambda ^{\mathsf{Sets}_{*}}_{Y}\circ \phi _{S^{0},Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ),\\ \lambda '_{Y} & = \lambda ^{\mathsf{Sets}_{*}}_{Y}\circ \psi _{S^{0},Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ). \end{align*}

    Postcomposing both sides with $\lambda ^{\mathsf{Sets}_{*},-1}_{Y}$ and then precomposing both sides with $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}$ gives

    \begin{align*} \lambda ^{\mathsf{Sets}_{*},-1}_{Y}\circ \lambda '_{Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ) & = \phi _{S^{0},Y},\\ \lambda ^{\mathsf{Sets}_{*},-1}_{Y}\circ \lambda '_{Y}\circ \webleft (\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}|\mathsf{Sets}}\otimes _{\mathsf{Sets}}\operatorname {\mathrm{id}}_{Y}\webright ) & = \psi _{S^{0},Y}, \end{align*}

    and thus we have

    \[ \phi _{S^{0},Y}=\psi _{S^{0},Y} \]

    for each $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$. Now, let $x\in X$ and consider the naturality diagrams

    for $\phi $ and $\psi $ with respect to the morphisms $\webleft [x\webright ]$ and $\operatorname {\mathrm{id}}_{Y}$. Having shown that $\phi _{S^{0},Y}=\psi _{S^{0},Y}$, we have

    \begin{align*} \phi _{X,Y}\webleft (x,y\webright ) & = \webleft [\phi _{X,Y}\circ \webleft (\webleft [x\webright ]\wedge \operatorname {\mathrm{id}}_{Y}\webright )\webright ]\webleft (1,y\webright )\\ & = \webleft [\webleft (\webleft [x\webright ]\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Y}\webright )\circ \phi _{S^{0},Y}\webright ]\webleft (1,y\webright )\\ & = \webleft [\webleft (\webleft [x\webright ]\otimes _{\mathsf{Sets}_{*}}\operatorname {\mathrm{id}}_{Y}\webright )\circ \psi _{S^{0},Y}\webright ]\webleft (1,y\webright )\\ & = \webleft [\psi _{X,Y}\circ \webleft (\webleft [x\webright ]\wedge \operatorname {\mathrm{id}}_{Y}\webright )\webright ]\webleft (1,y\webright )\\ & = \psi _{X,Y}\webleft (x,y\webright ) \end{align*}

    for each $\webleft (x,y\webright )\in X\wedge Y$. Therefore we have

    \[ \phi _{X,Y}=\psi _{X,Y} \]

    for each $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ and thus $\phi =\psi $, showing the isomorphism $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\times }$ to be unique.

    Corollary 7.5.10.1.2 ▶ A Second Universal Property for $\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright )$

    The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ of Proposition 7.5.9.1.1 is uniquely determined by the following requirements:

    1. 1.

      Two-Sided Preservation of Colimits. The tensor product

      \[ \otimes _{\mathsf{Sets}_{*}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*} \]

      of $\mathsf{Sets}_{*}$ preserves colimits separately in each variable.

    2. 2.

      The Unit Object Is $S^{0}$. We have $\mathbb {1}_{\mathsf{Sets}_{*}}\cong S^{0}$.

    More precisely, the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of Unresolved reference spanned by the symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ satisfying Item 1 and Item 2 is contractible.

    Proof of Corollary 7.5.10.1.2.

    Corollary 7.5.10.1.3 ▶ A Third Universal Property of the Smash Product of Pointed Sets

    The symmetric monoidal structure on the category $\mathsf{Sets}_{*}$ is the unique symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that the free pointed set functor

    \[ \webleft (-\webright )^{+} \colon \mathsf{Sets}\to \mathsf{Sets}_{*} \]

    admits a symmetric monoidal structure, i.e. the full subcategory of the category $\mathcal{M}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}_{*}\webright )$ of Unresolved reference spanned by the symmetric monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}_{*}}}}\mathsf{Sets}_{*}\right.$, $\otimes _{\mathsf{Sets}_{*}}$, $\mathbb {1}_{\mathsf{Sets}_{*}}$, $\lambda ^{\mathsf{Sets}_{*}}$, $\rho ^{\mathsf{Sets}_{*}}$, $\left.\sigma ^{\mathsf{Sets}_{*}}\right)$ with respect to which $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure is contractible.

    Proof of Corollary 7.5.10.1.3.

    Let $\webleft (\otimes _{\mathsf{Sets}_{*}},\mathbb {1}_{\mathsf{Sets}_{*}},\lambda ^{\mathsf{Sets}_{*}},\rho ^{\mathsf{Sets}_{*}},\sigma ^{\mathsf{Sets}_{*}}\webright )$ be a symmetric monoidal structure on $\mathsf{Sets}_{*}$ such that $\webleft (-\webright )^{+}$ admits a symmetric monoidal structure with respect to $\otimes _{\mathsf{Sets}_{*}}$ and $\wedge $. We have isomorphisms

    \begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (X^{-}\webright )^{+}\otimes _{\mathsf{Sets}_{*}}\webleft (Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\times Y^{-}\webright )^{+}\\ & \cong \webleft (X^{-}\webright )^{+}\wedge \webleft (Y^{-}\webright )^{+}\\ & \cong X\wedge Y, \end{align*}

    all natural in $X$ and $Y$. Now, since $\wedge $ preserves colimits in both variables and $\mathord {\otimes _{\mathsf{Sets}_{*}}}\cong \mathord {\wedge }$, it follows that $\otimes _{\mathsf{Sets}_{*}}$ also preserves colimits in both variables, so the result then follows from Corollary 7.5.10.1.2.

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