The Clowder Project

In detail, the moduli category of monoidal structures on $\mathcal{C}$ is the category ${\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)$ where:

• Objects. The objects of ${\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)$ are monoidal categories $(\phantom{{\lambda }^{\mathcal{C}}}\mathcal{C}$, ${\otimes }_{\mathcal{C}}$, ${\mathbb{1}}_{\mathcal{C}}$, ${\alpha }^{\mathcal{C}}$, ${\lambda }^{\mathcal{C}}$, ${\rho }^{\mathcal{C}})$ whose underlying category is $\mathcal{C}$.
• Morphisms. A morphism from $(\phantom{{\lambda }^{\mathcal{C}}}\mathcal{C}$, ${\otimes }_{\mathcal{C}}$, ${\mathbb{1}}_{\mathcal{C}}$, ${\alpha }^{\mathcal{C}}$, ${\lambda }^{\mathcal{C}}$, ${\rho }^{\mathcal{C}})$ to $(\phantom{{\lambda }^{\mathcal{C},\prime }}\mathcal{C}$, ${⊠}_{\mathcal{C}}$, ${\mathbb{1}}_{\mathcal{C}}^{\prime }$, ${\alpha }^{\mathcal{C},\prime }$, ${\lambda }^{\mathcal{C},\prime }$, ${\rho }^{\mathcal{C},\prime })$ is a strong monoidal functor structure
  $$\begin{array}{c}{\text{id}}_{\mathcal{C}}^{\otimes }:A{⊠}_{\mathcal{C}}B\stackrel{\sim }{⇢}A{\otimes }_{\mathcal{C}}B,\\ {\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes }:{\mathbb{1}}_{\mathcal{C}}^{\prime }\stackrel{\sim }{⇢}{\mathbb{1}}_{\mathcal{C}}\end{array}$$

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

• Identities. For each $M\stackrel{\text{def}}{=}(\phantom{{\lambda }^{\mathcal{C}}}\mathcal{C}$, ${\otimes }_{\mathcal{C}}$, ${\mathbb{1}}_{\mathcal{C}}$, ${\alpha }^{\mathcal{C}}$, ${\lambda }^{\mathcal{C}}$, ${\rho }^{\mathcal{C}})\in \text{Obj}\left({\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)\right)$, the unit map
  $${\mathbb{1}}_{M,M}^{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}:\text{pt}\to {\text{Hom}}_{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}(M,M)$$

  of ${\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)$ at $M$ is defined by

  $${\text{id}}_M^{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}\stackrel{\text{def}}{=}({\text{id}}_{\mathcal{C}}^{\otimes },{\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes }),$$

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

• Composition. For each $M,N,P\in \text{Obj}\left({\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)\right)$, the composition map
  $${\circ }_{M,N,P}^{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}:{\text{Hom}}_{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}(N,P)\times {\text{Hom}}_{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}(M,N)\to {\text{Hom}}_{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}(M,P)$$

  of ${\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)$ at $(M,N,P)$ is defined by

  $$({\text{id}}_{\mathcal{C}}^{\otimes ,\prime },{\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes ,\prime }){\circ }_{M,N,P}^{{\mathcal{M}}_{{\mathbb{E}}_1}\left(\mathcal{C}\right)}({\text{id}}_{\mathcal{C}}^{\otimes },{\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes })\stackrel{\text{def}}{=}({\text{id}}_{\mathcal{C}}^{\otimes ,\prime }\circ {\text{id}}_{\mathcal{C}}^{\otimes },{\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes ,\prime }\circ {\text{id}}_{\mathbb{1}|\mathcal{C}}^{\otimes }).$$