The moduli category of monoidal structures on $\mathcal{C}$ is the category $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ defined by
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Objects. The objects of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ are monoidal categories $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ whose underlying category is $\mathcal{C}$.
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Morphisms. A morphism from $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ to $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\right)$ is a strong monoidal functor structure
\begin{gather*} \operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}} \colon A\boxtimes _{\mathcal{C}}B \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A\otimes _{\mathcal{C}}B,\\ \operatorname {\mathrm{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 $\operatorname {\mathrm{id}}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ of $\mathcal{C}$.
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Identities. For each $M\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the unit map
\[ \mathbb {1}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,M} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,M\webright ) \]of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $M$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\webright ), \]where $\Big(\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big)$ is the identity monoidal functor of $\mathcal{C}$ of
.
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Composition. For each $M,N,P\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )\webright )$, the composition map
\[ \circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}\colon \operatorname {\mathrm{Hom}}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (N,P\webright )\times \operatorname {\mathrm{Hom}}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,N\webright )\to \operatorname {\mathrm{Hom}}_{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}\webleft (M,P\webright ) \]of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ at $\webleft (M,N,P\webright )$ is defined by
\[ \Big(\operatorname {\mathrm{id}}^{\otimes ,\prime }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\Big)\mathbin {\circ ^{\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )}_{M,N,P}}\Big(\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Big(\operatorname {\mathrm{id}}^{\otimes ,\prime }_{\mathcal{C}}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes ,\prime }_{\mathbb {1}|\mathcal{C}}\circ \operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\Big). \] -
1.
Naturality. For each pair $f\colon A\to X$ and $g\colon B\to Y$ of morphisms of $\mathcal{C}$, the diagram
commutes.
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Monoidality. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
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3.
Left Monoidal Unity. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
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4.
Right Monoidal Unity. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the diagram
commutes.
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1.
Extra Monoidality Conditions. Let $\big (\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\big )$ be a morphism of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ from $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ to $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\right)$.
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2.
Extra Monoidal Unity Constraints. Let $\big (\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathbb {1}|\mathcal{C}}\big )$ be a morphism of $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ from $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ to $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\right)$.
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(a)
The diagram
commutes.
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(b)
The diagram
commutes.
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(c)
The diagram
commutes.
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(d)
The diagram
commutes.
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3.
Mixed Associators. Let $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ and $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\right)$ be monoidal structures on $\mathcal{C}$ and let
\[ \operatorname {\mathrm{id}}^{\otimes }_{-_{1},-_{2}}\colon -_{1}\boxtimes _{\mathcal{C}}-_{2}\to -_{1}\otimes _{\mathcal{C}}-_{2} \]be a natural transformation.
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(a)
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 $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
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(b)
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 $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
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(c)
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 $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
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(a)
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1.
Proof of Item 1a: This follows from the naturality of $\operatorname {\mathrm{id}}^{\otimes }$ with respect to the morphisms $\operatorname {\mathrm{id}}^{\otimes }_{A,B}$ and $\operatorname {\mathrm{id}}_{C}$.
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2.
Proof of Item 1b: This follows from the naturality of $\operatorname {\mathrm{id}}^{\otimes }$ with respect to the morphisms $\operatorname {\mathrm{id}}_{A}$ and $\operatorname {\mathrm{id}}^{\otimes }_{B,C}$.
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Proof of Item 1a: Indeed, consider the diagram
whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathcal{C}}$;
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Subdiagram $\webleft (2\webright )$ commutes trivially;
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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
;
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Subdiagram $\webleft (4\webright )$ commutes by the right monoidal unity of $\webleft (\operatorname {\mathrm{id}}_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
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Proof of Item 1b: Indeed, consider the diagram
whose boundary diagram is the diagram whose commutativity we wish to prove. Since:
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Subdiagram $\webleft (1\webright )$ commutes by the naturality of $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathcal{C}}$;
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Subdiagram $\webleft (2\webright )$ commutes trivially;
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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
;
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Subdiagram $\webleft (4\webright )$ commutes by the left monoidal unity of $\webleft (\operatorname {\mathrm{id}}_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}},\operatorname {\mathrm{id}}^{\otimes }_{\mathcal{C}|\mathbb {1}}\webright )$;
so does the boundary diagram, and we are done.
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3.
Proof of Item 2c: Indeed, consider the diagram
Since:
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The boundary diagram commutes trivially;
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Subdiagram $\webleft (1\webright )$ commutes by Item 1b;
it follows that the diagram
commutes. But since $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}_{\mathcal{C}},\mathbb {1}'_{\mathcal{C}}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.
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Proof of Item 2d: Indeed, consider the diagram
Since:
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The boundary diagram commutes trivially;
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Subdiagram $\webleft (1\webright )$ commutes by Item 1a;
it follows that the diagram
commutes. But since $\operatorname {\mathrm{id}}^{\otimes ,-1}_{\mathbb {1}}$ is an isomorphism, it follows that the diagram $\webleft (\dagger \webright )$ also commutes, and we are done.
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Proof of Item 3a: We may partition the monoidality diagram for $\operatorname {\mathrm{id}}^{\otimes }$ of Item 2 of Remark 12.1.1.1.3 as follows:
Since:
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Subdiagram $\webleft (1\webright )$ commutes by Item 1a of Item 1.
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Subdiagram $\webleft (2\webright )$ commutes by assumption.
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Subdiagram $\webleft (3\webright )$ commutes by assumption.
it follows that the boundary diagram also commutes, i.e. $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
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Proof of Item 3b: We may partition the monoidality diagram for $\operatorname {\mathrm{id}}^{\otimes }$ of Item 2 of Remark 12.1.1.1.3 as follows:
Since:
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Subdiagram $\webleft (1\webright )$ commutes by assumption.
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Subdiagram $\webleft (2\webright )$ commutes by assumption.
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Subdiagram $\webleft (3\webright )$ commutes by Item 1b of Item 1.
it follows that the boundary diagram also commutes, i.e. $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
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3.
Proof of Item 3c: We may partition the monoidality diagram for $\operatorname {\mathrm{id}}^{\otimes }$ of Item 2 of Remark 12.1.1.1.3 as follows:
Since subdiagrams (1) and (2) commute by assumption, it follows that the boundary diagram also commutes, i.e. $\operatorname {\mathrm{id}}^{\otimes }$ satisfies the monoidality condition of Item 2 of Remark 12.1.1.1.3.
12.1.1 The Moduli Category of Monoidal Structures on a Category
Let $\mathcal{C}$ be a category.
In detail, the moduli category of monoidal structures on $\mathcal{C}$ is the category $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ where:
In particular, a morphism in $\mathcal{M}_{\mathbb {E}_{1}}\webleft (\mathcal{C}\webright )$ from $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C}}}}\mathcal{C}\right.$, $\otimes _{\mathcal{C}}$, $\mathbb {1}_{\mathcal{C}}$, $\alpha ^{\mathcal{C}}$, $\lambda ^{\mathcal{C}}$, $\left.\rho ^{\mathcal{C}}\right)$ to $\left(\phantom{\mathrlap {\lambda ^{\mathcal{C},\prime }}}\mathcal{C}\right.$, $\boxtimes _{\mathcal{C}}$, $\mathbb {1}'_{\mathcal{C}}$, $\alpha ^{\mathcal{C},\prime }$, $\lambda ^{\mathcal{C},\prime }$, $\left.\rho ^{\mathcal{C},\prime }\right)$ satisfies the following conditions:
Let $\mathcal{C}$ be a category.
Proof of Proposition 12.1.1.1.4.
Item 1: Extra Monoidality Conditions
We claim that Item 1a and Item 1b are indeed true:
This finishes the proof.
Item 2: Extra Monoidal Unity Constraints
We claim that Item 2a and Item 2b are indeed true:
This finishes the proof.
Item 3: Mixed Associators
We claim that Item 3a, Item 3b, and Item 3c are indeed true:
This finishes the proof.
