Proposition 10.1.1.1.4 (01UR): Properties of the Moduli Category of Monoidal Structures on a Category

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

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.
Extra Monoidal Unity Constraints. 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.
3. The diagram
  
  commutes.
4. The diagram
  
  commutes.
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
\[ \text{id}^{\otimes }_{-_{1},-_{2}}\colon -_{1}\boxtimes _{\mathcal{C}}-_{2}\to -_{1}\otimes _{\mathcal{C}}-_{2} \]

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.

Item 1: Extra Monoidality Conditions

We claim that Item (a) and Item (b) are indeed true:

Proof of Item (a): This follows from the naturality of $\text{id}^{\otimes }$ with respect to the morphisms $\text{id}^{\otimes }_{A,B}$ and $\text{id}_{C}$.
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: Extra Monoidal Unity Constraints

We claim that Item (a) and Item (b) are indeed true:

Proof of Item (a): Indeed, consider the diagram

whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
- Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathcal{C}}$;
- Subdiagram $\webleft (2\webright )$ commutes trivially;
- Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\lambda ^{\mathcal{C}}$, where the equality $\rho ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}=\lambda ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}$ comes from ;
- Subdiagram $\webleft (4\webright )$ commutes by the right monoidal unity of $\webleft (\text{id}_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
Proof of Item (b): Indeed, consider the diagram

whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
- Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathcal{C}}$;
- Subdiagram $\webleft (2\webright )$ commutes trivially;
- Subdiagram $\webleft (3\webright )$ commutes by the naturality of $\rho ^{\mathcal{C}}$, where the equality $\rho ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}=\lambda ^{\mathcal{C}}_{\mathbb {1}_{\mathcal{C}}}$ comes from ;
- Subdiagram $\webleft (4\webright )$ commutes by the left monoidal unity of $\webleft (\text{id}_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}},\text{id}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
Proof of Item (c): Indeed, consider the diagram

Since:
- The boundary diagram commutes trivially;
- Subdiagram $\webleft (1\webright )$ commutes by Item (b);
it follows that the diagram

commutes. But since $\text{id}^{\otimes ,-1}_{\mathbb {1}_{\mathcal{C}},\mathbb {1}'_{\mathcal{C}}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.
Proof of Item (d): Indeed, consider the diagram

Since:
- The boundary diagram commutes trivially;
- Subdiagram $\webleft (1\webright )$ commutes by Item (a);
it follows that the diagram

commutes. But since $\text{id}^{\otimes ,-1}_{\mathbb {1}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.

This finishes the proof.

Item 3: Mixed Associators

We claim that Item (a), Item (b), and Item (c) are indeed true:

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 $\webleft (1\webright )$ commutes by Item (a) of Item 1.
- Subdiagram $\webleft (2\webright )$ commutes by assumption.
- Subdiagram $\webleft (3\webright )$ 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.
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 $\webleft (1\webright )$ commutes by assumption.
- Subdiagram $\webleft (2\webright )$ commutes by assumption.
- Subdiagram $\webleft (3\webright )$ 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.
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.