10.1.1 The Moduli Category of Monoidal Structures on a Category

Let $\mathcal{C}$ be a category.

Definition 10.1.1.1.1 ▶ The Moduli Category of Monoidal Structures on a Category

The moduli category of monoidal structures on $\mathcal{C}$ is the category $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ defined by

Remark 10.1.1.1.2 ▶ Unwinding Definition 10.1.1.1.1, I

In detail, the moduli category of monoidal structures on $\mathcal{C}$ is the category $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ where:

  • Objects. The objects of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ are monoidal categories $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ whose underlying category is $\mathcal{C}$.
  • Morphisms. A morphism from $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ to $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$ is a monoidal functor structure

    \begin{gather*} \text{id}^{\otimes }_{\mathcal{C}} \colon A\otimes _{\mathcal{C}}B \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A\boxtimes _{\mathcal{C}}B,\\ \text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}} \colon \mathbb {1}_{\mathcal{C}} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathbb {1}’_{\mathcal{C}} \end{gather*}

    on the identity functor $\text{id}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ of $\mathcal{C}$.

  • Identities. For each $M\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)\in \text{Obj}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the unit map

    \[ \mathbb {1}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,M} \colon \text{pt}\to \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,M\webright ) \]

    of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $M$ is defined by

    \[ \text{id}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\text{id}_{\otimes _{\mathcal{C}}},\text{id}_{\mathbb {1}_{\mathcal{C}}}\webright ), \]

    where $\Big(\text{id}_{\otimes _{\mathcal{C}}},\text{id}_{\mathbb {1}_{\mathcal{C}}}\Big)$ is the identity monoidal functor of .

  • Composition. For each $M,N,P\in \text{Obj}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the composition map

    \[ \circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}\colon \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (N,P\webright )\times \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,N\webright )\to \textup{Hom}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,P\webright ) \]

    of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $\webleft (M,N,P\webright )$ is defined by

    \[ \Big(\text{id}^{\otimes ,\prime }_{\mathcal{C}},\text{id}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\Big)\mathbin {\circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}}\Big(\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Big(\text{id}^{\otimes ,\prime }_{\mathcal{C}}\circ \text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\circ \text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big). \]

Remark 10.1.1.1.3 ▶ Unwinding Definition 10.1.1.1.1, II

In particular, a morphism in $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ from $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ to $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$ satisfies the following conditions:

  1. Naturality. For each pair $f\colon A\to X$ and $g\colon B\to Y$ of morphisms of $\mathcal{C}$, the diagram

    commutes.

  2. Monoidality. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram

    commutes.

  3. Left Monoidal Unity. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram

    commutes.

  4. Right Monoidal Unity. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram

    commutes.

Proposition 10.1.1.1.4 ▶ Properties of the Moduli Category of Monoidal Structures on a Category

Let $\mathcal{C}$ be a category.

  1. Extra Monoidality Conditions. Let $\big (\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathbb {1}|\mathcal{C}}\big )$ be a morphism of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ from $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ to $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$.
    1. The diagram

      commutes.

    2. The diagram

      commutes.

  2. Mixed Associators. Let $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\webright)$ and $\webleft(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\webright)$ be monoidal structures on $\mathcal{C}$ and let
    \[ \text{id}^{\otimes }_{-_{1},-_{2}}\colon -_{1}\otimes _{\mathcal{C}}-_{2}\to -_{1}\boxtimes _{\mathcal{C}}-_{2} \]

    be a natural transformation.

    1. If there exists a natural transformation
      \[ \alpha ^{\otimes }_{A,B,C}\colon \webleft (A\otimes _{\mathcal{C}}B\webright )\boxtimes _{\mathcal{C}}C\to A\otimes _{\mathcal{C}}\webleft (B\boxtimes _{\mathcal{C}}C\webright ) \]

      making the diagrams

      and

      commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

    2. If there exists a natural transformation
      \[ \alpha ^{\boxtimes }_{A,B,C}\colon \webleft (A\boxtimes _{\mathcal{C}}B\webright )\otimes _{\mathcal{C}}C\to A\boxtimes _{\mathcal{C}}\webleft (B\otimes _{\mathcal{C}}C\webright ) \]

      making the diagrams

      and

      commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

    3. If there exists a natural transformation
      \[ \alpha ^{\boxtimes ,\otimes }_{A,B,C}\colon \webleft (A\boxtimes _{\mathcal{C}}B\webright )\otimes _{\mathcal{C}}C\to A\otimes _{\mathcal{C}}\webleft (B\boxtimes _{\mathcal{C}}C\webright ) \]

      making the diagrams

      and

      commute, then the natural transformation $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

Proof of Proposition 10.1.1.1.4.
Item 1: Extra Monoidality Conditions
We claim that Item (a) and Item (b) are indeed true:
  1. Proof of Item (a): This follows from the naturality of $\text{id}^{\otimes }$ with respect to the morphisms $\text{id}^{\otimes }_{A,B}$ and $\text{id}_{C}$.
  2. Proof of Item (b): This follows from the naturality of $\text{id}^{\otimes }$ with respect to the morphisms $\text{id}_{A}$ and $\text{id}^{\otimes }_{B,C}$.

This finishes the proof.

Item 2: Mixed Associators
We claim that Item (a), Item (b), and Item (c) are indeed true:
  1. Proof of Item (a): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:

    Since:

    • Subdiagram (1) commutes by Item (a) of Item 1.
    • Subdiagram (2) commutes by assumption.
    • Subdiagram (3) commutes by assumption.
    it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

  2. Proof of Item (b): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:

    Since:

    • Subdiagram (1) commutes by assumption.
    • Subdiagram (2) commutes by assumption.
    • Subdiagram (3) commutes by Item (b) of Item 1.
    it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

  3. Proof of Item (c): We may partition the monoidality diagram for $\text{id}^{\otimes }$ of Item 2 of Remark 10.1.1.1.3 as follows:

    Since subdiagrams (1) and (2) commute by assumption, it follows that the boundary diagram also commutes, i.e. $\text{id}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 10.1.1.1.3.

This finishes the proof.

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