\mathcal{O}_G \xrightarrow{\sim} \Delta^\ast_{GF,G}\mathcal{O}_{GF}  \xrightarrow{\sim} \Delta^\ast_{GF,G}  \beta^\ast \mathcal{O}_{FG}  \xrightarrow{\sim} \Delta^\ast_{GF,G} \beta^\ast  \upmu^\ast_{FG,F}  \mathcal{O}_F
\]

\[\xrightarrow{\Delta^\ast_{GF,G}  (\beta^\ast(\upmu^\ast_{FG,F}  (s)))} \Delta^\ast_{GF,G}  \beta^\ast \upmu^\ast_{FG,F}  M \xrightarrow{{\text{B}_{F,G,A}}_M}  \mu^\ast_{G,A}\Delta^\ast_{F,A} M

The first map is given on stalks by

\[
\text{f} \mapsto 1 \otimes \text{f} \mapsto  1 \otimes 1 \otimes \text{f} \mapsto   s(\text{1}) \otimes 1 \otimes \text{f}
\]

and the second map by

\[
\text{f} \mapsto 1 \otimes \text{f} \mapsto  1 \otimes 1 \otimes \text{f} \mapsto 1 \otimes  1 \otimes 1 \otimes \text{f}\mapsto s(\text{1}) \otimes1 \otimes 1 \otimes \text{f}  \mapsto s(\text{1})  \otimes 1 \otimes \text{f}
.\qedhere \]

§ INVERTIBLE SHEAVES AND BOOLEAN FUNCTIONS

We now study an indexing of sheaves of modules which has essential image the invertible sheaves. Invertible sheaves are often constructed as subsheaves of the sheaf of rational functions, and may be indexed by torus invariant Cartier divisors. However, we require a slightly different approach in order to expose the bimonoid structure of invertible sheaves.

§.§ The Bimonoid of Boolean Functions

Let $[ I,\text{2} ]$ denote the set of all subsets $A\subseteq I$. We have the Joyal-theoretic cospecies of integer-valued Boolean functions $\textsf{BF}$, given by

\[
\textsf{BF}[I]:= \big \{  \text{functions}\ z: [ I,\text{2} ] \to \bZ  \ \big |\ z(\emptyset)=0 \big \}
,\qquad   \textsf{BF}[\sigma](z)\, (A) :=  z( A')

In particular, for each integer point $h=(h_\text{i})_{\text{i}\in I}\in \bZ I$ we have the Boolean function $z_h$ given by

\[
z_h(A) : =  \la h, \lambda_A \ra =  \sum_{\text{i}\in A} h_\text{i}

We let $\textsf{BF}[I]$ be a category by including a single morphism $z_1\to z_2$ whenever $z_2=z_1 + z_h$ for some $h\in \bZ I$. This is an equivalence relation.

We call $\text{hei}(z) := z(I)$ the height of $z$. For $z=(z_1,\dots,z_k)\in \textsf{BF}[F]$, height is a $k$-tuple

\[
\text{hei}(z_1,\dots,z_k):= \big(\text{hei}(z_1),\dots,\text{hei}(z_k)\big)

We now give $\textsf{BF}$ the structure of a bimonoid, following [1]. Given Boolean functions $z_1\in \textsf{BF}[S]$ and $z_2\in \textsf{BF}[T]$, let $(z_1|z_2)\in \textsf{BF}[I]$ be the Boolean function given by

\[
(z_1|z_2)(A):=z_1( A \cap S ) + z_2(  A \cap T)

Given a Boolean function $z\in \textsf{BF}[I]$, let $z\talloblong_S\in \textsf{BF}[S]$ and $z \! \!  \fatslash_{\, T}\in \textsf{BF}[T]$ be the Boolean functions given by

\[
z\! \talloblong_S(A):= z(A)
\qquad \text{and} \qquad
z \! \! \!  \fatslash_{\, T}(A):=  z(A \sqcup S) -z(S)

Then $\textsf{BF}$ is a Joyal-theoretic bimonoid in $\textsf{Cat}$-valued cospecies, with multiplication and comultiplication given by

\[
\mu_{S,T}(z_1,z_2) :=(z_1|z_2)
\qquad \text{and} \qquad
\Delta_{S,T} (z) := (z\talloblong_S, z \! \!  \fatslash_{\, T})
\]

respectively. For the higher comultiplication, we denote e.g.

\[
(z\talloblong_S , z\talloblong_T , z\talloblong_U):= \Delta_{(S,T,U)} (z)

§.§ Invertible Sheaves From Cartier Divisors

We also have the Joyal-theoretic cospecies $\bZ^{[ -;\text{2} ]}$, given by

\[
\bZ^{[ I;\text{2} ]}:= \big\{ \text{functions}\ w: [ I;\text{2} ] \to \bZ  \big\}
\sigma(w)\big ((S,T)\big ):= w\big( (S',T')  \big)

The functions $w\in \bZ^{[ I;\text{2} ]}$ are in one-to-one correspondence with torus invariant Cartier divisors $D_w$ on permutohedral space $\bC\Sigma^I$. Recall the invertible sheaf $\mathcal{O}_w\in \text{Mod}_I$ associated to $D_w$ is the submodule of the sheaf of rational functions given by

\[
\mathcal{O}_w(U) :=      \big \{g\in   \mathcal{K}_I
\ \big |\
g=0 \text{ or }    \text{div}g +D_w |_U \geq 0   \big \}

See e.g. <cit.>. The essential image of the indexing $w\mapsto \mathcal{O}_w$ is all the invertible sheaves.

We have the inclusion of height zero Boolean functions

\[
\bZ^{[-;2]}\hookrightarrow \textsf{BF}
w\,  \mapsto\,
\begin{cases}
\big ( A\mapsto w(  A, B) \big  ), \quad \text{for $A\neq I,\emptyset$}     \\
(I\mapsto 0) \\
(\emptyset \mapsto 0)
\end{cases}
\]

where $B=I-A$. The idea is to now generalize this construction of invertible sheaves to all of $\textsf{BF}$.

§.§ Affine Semisimple Flats

We give a geometric interpretation of the algebraic structure of $\textsf{BF}$, analogous to what we found for augmented preposets $\textsf{O}_\bullet$. First, we need an analog of the monoid $\text{T}^\vee$.

For $a \in \bZ$, let $\text{T}^\vee_{I, a }$ denote the `sum equal to $a$' hyperplane of $\bR I$, given by

\[
\text{T}^\vee_{I, a } =
\Big \{
h\in \bR I  \ \Big | \  \sum_{\text{i}\in I} h_\text{i} = a
\Big \}

For $F=(S_1,\dots, S_k)$ a composition of $I$ and a $k$-tuple of integers $(a_1 , \dots , a_k) \in \bZ^k$, we have the affine semisimple flat $\textsf{T}^\vee_{F,(a_1 , \dots , a_k)}$, given by

\[
\textsf{T}^\vee_{F,(a_1 , \dots , a_k)} :=  \Big \{    h\in \bR I \ \Big | \   \sum_{\text{i}\in S_i} h_\text{i}   = a_i    \quad \text{for all} \quad 1\leq i \leq k    \Big \}

A key fact is that semisimple flats factorize; we have the bijective map $\eta_{F,(a_1 , \dots , a_k)}$ given by

\[
\eta_{F,(a_1 , \dots , a_k)}:
\text{T}^\vee_{S_1, a_1 }  \times   \dots \times \text{T}^\vee_{S_k, a_k }
\xrightarrow{\sim}     \textsf{T}^\vee_{ F,(a_1 , \dots , a_k) }
\big ( (h_\text{i})_{\text{i}\in S_1} ,\dots ,(h_\text{i})_{\text{i} \in S_k} \big )  \mapsto  (h_\text{i})_{\text{i}\in I}

For compositions $F, G$ with $G\leq F$, let

\[
F=(S_1,\dots,  S_k)\qquad \text{and} \qquad G=(T_1,\dots,  T_l)
\]

and let $1\leq \mathtt{i}_j \leq k$ be the integer such that $S_{\mathtt{i}_j}$ is the $\mathtt{i}$th lump of $F|_{T_j}$, $1\leq j \leq l$. Let $k(j)$ be the length of $F|_{T_j}$. Then we have the function ${\mu_{F,G}}_{(  a_1,\dots,a_k )}$ the inclusion of semisimple flats,

\[
{\mu_{F,G}}_{(  a_1,\dots,a_k )}   : \textsf{T}^\vee_{F,(  a_1,\dots,a_k )} \hookrightarrow \textsf{T}^\vee_{G,(  a_{1_1}+\,  \cdots\,  + a_{k(1)_{1}}  \,  , \dots \dots   ,  \,  a_{1_l}+\, \cdots \, + a_{k(l)_l} )}
h\mapsto h

We have the function the composition $\text{m}_{F,(a_1 , \dots , a_k)}:={\mu_{F,(I)}}_{(a_1 , \dots , a_k)} \circ {\eta_F}_{(a_1 , \dots , a_k)}$, which is given by

\[
\text{m}_{F,(a_1 , \dots , a_k)}:
\text{T}^\vee_{S_1, a_1}  \times   \dots \times \text{T}^\vee_{S_k, a_k}
\to  \text{T}^\vee_{I,a_1 + \dots + a_k}
\big ( (h_\text{i})_{\text{i}\in S_1} ,\dots ,(h_\text{i})_{\text{i} \in S_k} \big )  \mapsto  (h_\text{i})_{\text{i}\in I}

In fact, $(F,a)\mapsto \textsf{T}^\vee_{F,a}$ defines a $\bZ$-graded set cospecies, where $\bZ$ denotes the constant cospecies $F\mapsto \bZ$. Moreover, this cospecies is a $\bZ$-graded monoid, with multiplication the inclusion of semisimple flats. Then $\text{m}_{F,(a_1 , \dots , a_k)}$ is the multiplication ${\mu_{F,(I)}}_{(a_1 , \dots , a_k)}$ of its Joyalization.

§.§ Plate Arrangements

To each Boolean function $z\in \textsf{BF}[I]$ we associate the halfspace arrangement in $\text{T}_{I,\text{hei}(z)}^\vee$ which consists of the halfspaces

\begin{align*}
[[A,B]]_z  :=&\  \big \{     h\in \text{T}^\vee_{I, \text{hei}(z)}  \ \big | \       \la h, \lambda_{A}  \ra \leq z(A)  \big  \}  \\
=&\  \big \{     h\in \text{T}^\vee_{I, \text{hei}(z)}  \ \big | \        \la h, \lambda_{B}  \ra \geq z(I)-z(A)   \big  \}
\end{align*}

where $(A,B)\in [I; 2]$. More generally, we have the associated plate arrangement, which consists of the plates

\[
[[ H ]]_z:=  \bigcap_{(A,B) \leq H}  [[A,B]]_z
\]

\[
=\ \big \{     h\in \text{T}^\vee_{I, \text{hei}(z)}       \ \big | \      \la h, \lambda_{\overline{A}_j}  \ra \leq z\big (\overline{A}_j\big ), \quad 0< j< \ell     \big \}
\]

\[
=\ \big \{     h\in \text{T}^\vee_{I, \text{hei}(z)}       \ \big | \      \la h, \lambda_{\overline{A}_j}  \ra \leq z\big (\overline{A}_j\big ), \quad 0\leq j\leq \ell     \big \}
\]

where $H= (A_1,\dots, A_\ell) \in \Sigma[I]$ and

\[
\overline{A}_j:=A_1\sqcup\dots \sqcup  A_j, \qquad  0 \leq j \leq \ell

Plates were introduced in [26], and are studied in [11], [7]. A set of the form $\overline{A}_j$, $0\leq j \leq \ell$, is called an initial segment of $H$. In particular, we have $\overline{A}_0=\emptyset$ and $\overline{A}_\ell=I$. Thus, as a region $[[ H ]]_z\subset \bR I$, the plate is given by the inequalities

\begin{equation}\label{eq:inequal}
\la h, \lambda_{\overline{A}_j}  \ra \leq z\big (\overline{A}_j\big )
\end{equation}

for $0< j< \ell$, together with the equality for the ambient hyperplane

\begin{equation}\label{eq:equal}
\la h, \lambda_{I}  \ra =   z(I)
\end{equation}

If we let $j=0$ in (eq:inequal) we obtain the trivial inequality

\[
0= \la h, \lambda_{  \emptyset }  \ra \leq   z(\emptyset)=0
\]

and if we let $j=\ell$ in (eq:inequal) we obtain

\[
\la h, \lambda_{I}  \ra \leq   z(I)
\]

which is implied by (eq:equal).

The plate $[[ H ]]_z$ is a module of the monoid $\sigma_H^o$, with the action given by addition of vectors in $\bR I$. If $z_2=z_1 + z_{h}$, we have the bijection

\[
[[H ]]_{z_1} \xrightarrow{\sim} [[H ]]_{z_2}
(-)\, \mapsto \,  (-)+h

The maximum affine subspace of the plate $[[H]]_z$ is the semisimple flat

\[
\text{max aff} \big ( [[H]]_z \big ) = \textsf{T}^\vee_{H, \text{hei} (\Delta_H(z)  ) }

The maximum affine subspace of the plate $	[[ H ]]_z$ is obtained by replacing inequalities with equalities in (eq:inequal),

\[
\text{max aff} \big ( [[H]]_z \big ) =\big \{     h\in \text{T}^\vee_{I, \text{hei}(z)}       \ \big | \      \la h, \lambda_{\overline{A}_j}  \ra = z\big (\overline{A}_j\big ), \quad 0< j< \ell     \big \}.\]

Then notice this is just

\[
=  \textsf{T}^\vee_{H,(  z({A}_1),  z(\overline{{A}}_2- \overline{{A}}_1   ),      \dots,  z(\overline{{A}}_{\ell-1}- \overline{{A}}_{\ell-2}),  z(I- \overline{{A}}_{\ell-1}   )  )}
=  \textsf{T}^\vee_{H, \text{hei} (\Delta_H(z)  ) }
.\qedhere \]

For $F=(S_1,\dots, S_k)$ a composition of $I$, given an $F$-tuple of Boolean functions $z=(z_1,\dots, z_k) \in \textsf{BF}[F]$ and a composition $H\in \Sigma[I]$, the multiplication $\text{m}_{F,\text{hei}(z)}$ restricts to an embedding

\[
\text{m}_{F,\text{hei}(z)}:
[[    H|_{S_1}  ]]_{z_1}  \times \dots \times       [[    H|_{S_k}  ]]_{z_k}  \hookrightarrow   [[    H  ]]_{(z_1 | \dots | z_k)}
\]

with image given by

\[
\bigcap_{(A,B)\leq (   H|_{S_1} | \dots |    H|_{S_k}   )}   [[    A,B  ]]_{(z_1 | \dots | z_k)}

Let $H=(A_1,\dots, A_\ell)$, and let

\[
A_{j,i} =  S_i\cap A_j
\overline {A}_{j,i} =  A_{1,i}     \sqcup \dots \sqcup   A_{j,i}

We have

\[
H|_{S_i}=     (   A_{1,i}, \dots , A_{\ell,i} )_+

The plate $[[    H|_{S_i}  ]]_{z_i}  \subset \bR S_i$ is the region satisfying the inequalities

\begin{equation}\label{eq:1}
\la h, \lambda_{\overline{A}_{j,i}}  \ra \leq z_i(\overline{A}_{j,i})
\end{equation}

for $0< j <  \ell$, together with the equality for the ambient space

\begin{equation}\label{eq:2}
\la h, \lambda_{ S_i }  \ra = z_i(S_i)

The product $ [[    H|_{S_1}  ]]_{z_1}  \times \dots \times       [[    H|_{S_k}  ]]_{z_k}$ is the region satisfying these equalities for all $1\leq i \leq k$. On the other hand, the plate $\bigcap_{(A,B)\leq (   H|_{S_1} | \dots |    H|_{S_k}   )}   [[    A,B  ]]_{(z_1 | \dots | z_k)} \subset \bR I$ is the region satisfying the inequalities

\begin{equation}\label{eq:3}
\la h, \lambda_{ A }  \ra  \leq  (z_1| \dots | z_k) (A ) =     z_1 ( A\cap S_1 ) + \dots +z_k ( A\cap S_k )
\end{equation}

where $(A,B)\leq ( H|_{S_1} | \dots |    H|_{S_k}   )$, together with the equality for the ambient space

\begin{equation}\label{eq:4}
\la   h, \lambda_I \ra = (z_1| \dots | z_k)(I)= z_1(S_1) +\dots  + z_k(S_k)

Let $\mathcal{A}_i:= A\cap S_i$, so that $A=\mathcal{A}_1 \sqcup \dots \sqcup \mathcal{A}_k$ where $\mathcal{A}_i$ is an initial segment of $H|_{S_i}$. Then (eq:1) & (eq:2)$\implies$(eq:3) since

\[
\la h, \lambda_{A}  \ra =  \la h, \lambda_{\mathcal{A}_1}\ra + \dots + \la h, \lambda_{\mathcal{A}_k} \ra    \leq z_1(\mathcal{A}_1) +\dots + z_k(\mathcal{A}_k)

Clearly (eq:2)$\implies$(eq:4). We have (eq:3)$\implies$(eq:1) by setting $A=\overline{A}_{j,i}$ in (eq:3). We prove (eq:3) & (eq:4) $\implies$(eq:2) in two parts. First, set $A=S_i$ in (eq:3) to give

\[
\la h, \lambda_{ S_i }  \ra \leq z_i(S_i)

Second, set $A=I-S_i$ in (eq:3) to give[ where the hat $\hat{z}_i(S_i)$ means the term $z_i(S_i)$ is not included in the sum]

\begin{equation}\label{eq:5}
\la h, \lambda_{ I-S_i }  \ra \leq z_1(S_1) + \dots +  \hat{z}_i(S_i)  +   \dots + z_k(S_k)

\[
\la h, \lambda_{ S_i }  \ra  =
\lefteqn{\overbrace{\phantom{          \la h, \lambda_{ I}  \ra  - \la h, \lambda_{ I-S_i }  \ra    =  z_1(S_1) +   \dots + z_k(S_k)- \la h, \lambda_{ I-S_i } \ra                    }}^{\text{by \textcolor{blue}{(\refeqq{eq:4})}}}}
\la h, \lambda_{ I}  \ra  - \la h, \lambda_{ I-S_i }  \ra    =       \underbrace{     z_1(S_1) +   \dots + z_k(S_k)- \la h, \lambda_{ I-S_i } \ra  \geq   z_i(S_i)            }_{\text{by \textcolor{blue}{(\refeqq{eq:5})}}}

Thus, we have shown $\la h, \lambda_{ S_i }  \ra \leq z_i(S_i) $ and $\la h, \lambda_{ S_i }  \ra \geq z_i(S_i) $, and so $\la h, \lambda_{ S_i }  \ra = z_i(S_i) $. This completes the proof of the image.

Finally, we need to show the image is contained in $  [[    H  ]]_{(z_1 | \dots | z_k)} $. We have

\[
H \leq (   H|_{S_1} | \dots |    H|_{S_k}   )
\]

and so

\[
(A,B)\leq H \quad \implies \quad (A,B) \leq (   H|_{S_1} | \dots |    H|_{S_k}   )

\[
\bigcap_{(A,B)\leq (   H|_{S_1} | \dots |    H|_{S_k}   )}   [[    A,B  ]]_{(z_1 | \dots | z_k)} \ \subseteq \   [[    H  ]]_{(z_1 | \dots | z_k)}
.\qedhere \]

Given $(S,T)\in [I;2]$, the $(S,T)$-face of the plate $[[H]]_z$ is the region of $[[H]]_z$ on which the function $\lambda_{S}$ is maximized. Given $F\in \Sigma[I]$, the $F$-face of $[[H]]_z$ is the intersection of all the $(S,T)$-faces for $(S,T)\leq F$, i.e. it is the region on which $\lambda_{S}$ is maximized for all $(S,T)\leq F$.

If $F\leq H$, then the $F$-face of the plate $[[ H ]]_z  $ is given by the intersection

\[
[[H]]_z   \cap      \textsf{T}^\vee_{F, \text{hei}(\Delta_F(z))  }

If $F\nleq H$, then $[[ H ]]_z  $ does not have an $F$-face.

\[
H= (A_1,\dots, A_\ell)

First consider the case $F=(S,T)$. If $(S,T)\leq H$, then $\lambda_{S}\leq z(S)$ on $[[ H ]]_z  $. But $\lambda_{S}=z(S)$ on $\textsf{T}^\vee_{(S,T), \text{hei}(\Delta_{S,T}(z))  } $ by definition, and $[[ H ]]_z      \cap  \textsf{T}^\vee_{(S,T), \text{hei}(\Delta_{S,T}(z))  } $ is nonempty because $ \textsf{T}^\vee_{(S,T), \text{hei}(\Delta_{S,T}(z))  } $ contains the maximal affine subspace of $[[ H ]]_z $.

For generic $F$, the result then follows since

\[
\textsf{T}^\vee_{F, \text{hei}(\Delta_F(z))  } = \text{max aff} \big ([[F]]_z\big )
=\text{max aff} \big (   \bigcap_{(S,T)\leq F} [[S,T]]_z             \big )
\]

\[
=    \bigcap_{(S,T)\leq F}   \text{max aff} \big ( [[S,T]]_z \big )
=    \bigcap_{(S,T)\leq F}   \textsf{T}^\vee_{(S,T), \text{hei}(\Delta_{S,T}(z))  }

If $(S,T)\nleq H$, then $(t,s)\in H$ for some $s\in S$ and $t\in T$. Therefore $h_{s t}\in \sigma^o_H$, and so $\lambda_{S}$ obtains arbitrary large values on the $\sigma^o_H$-module $[[H]]_z$. If $F\nleq H$, then there exists $(S,T)\leq F$ with $(S,T)\nleq H$.

For $F=(S_1,\dots, S_k)$ a composition of $I$, given an $F$-tuple of compositions $K=(K_1,\dots, K_k)\in \mathsf{\Sigma}[F]$ and a Boolean function $z\in \textsf{BF}[I]$, the function $\text{m}_{F,\text{hei}(\Delta_F(z))}$ restricts to an embedding of cones

\[
[[ K_1    ]]_{z\talloblong_{S_1}}
\times \dots \times
[[ K_k   ]]_{z\talloblong_{S_k}  }  \hookrightarrow   [[ K_1 ; \dots ; K_k ]]_z
\]

with image the $F$-face of $[[ K_1 ; \dots ; K_k ]]_z$.

In light of <ref>, this amount to showing that the image is given by the intersection

\[
[[ K_1 ; \cdots ; K_k]]_z   \cap      \textsf{T}^\vee_{F, \text{hei}(\Delta_F(z))  }

\[
K_{i}   = (  B_{1,i},\dots, B_{l(i),i} )
\qquad \text{and} \qquad
\overline{B}_{j,i} :=   B_{1,i} \sqcup \dots \sqcup   B_{j,i}
\]

for $1\leq j \leq l(i)$. Then the intersection $[[ K_1 ; \cdots ; K_k ]]_z\cap \textsf{T}^\vee_{F,\text{hei}(\Delta_F(z))} \subset \bR I$ is the region given by the plate inequalities

\begin{equation}\label{eq:11}
\la h, \lambda_{  \overline{S}_{i-1}\sqcup   \overline{B}_{j,i}       }  \ra \leq z(\overline{S}_{i-1}\sqcup \overline{B}_{j,i}), \qquad 0\leq j < l(i)
\end{equation}

for $1\leq i \leq k$ (although the equality for $i=1$, $j=0$ is trivial), together with the equalities

\begin{equation}\label{eq:33}
\la h, \lambda_{S_i}  \ra =  \text{hei}(z\talloblong_{S_i})  =  z(\overline{S}_i) - z(\overline{S}_{i-1})    \qquad \text{for} \quad 1\leq i \leq k
\end{equation}

On the other hand, $[[ K_i    ]]_{z\talloblong_{S_i}} \subset \bR S_i$ is the region given by the inequalities

\begin{equation}\label{eq:55}
\la h, \lambda_{\overline{B}_{j,i} }  \ra \leq   z\! \talloblong_{S_i}(\overline{B}_{j,i})  =   z(\overline{S}_{i-1} \sqcup    \overline{B}_{j,i}  )  -  z(\overline{S}_{i-1})
\end{equation}

for $0< j < l(i)$, together with the equality for the ambient space

\begin{equation}\label{eq:66}
\la h, \lambda_{ S_i      }  \ra   = z\talloblong_{S_i} (S_i)  = z(\overline{S}_i) -   z(\overline{S}_{i-1})
\end{equation}

The product $[[ K_1    ]]_{z\talloblong_{S_1}} \times \dots \times[[ K_k   ]]_{z\talloblong_{S_k}  }$ is the region satisfying these equalities for all $1\leq i \leq k$.

Clearly $\textcolor{blue}{(\refeqq{eq:33})}$ $\iff$ $\textcolor{blue}{(\refeqq{eq:66})}$. For $\textcolor{blue}{(\refeqq{eq:11})}$ & $\textcolor{blue}{(\refeqq{eq:33})}$ $\implies$ $\textcolor{blue}{(\refeqq{eq:55})}$, we have

\[
\la h, \lambda_{\overline{B}_{j,i}   }  \ra
\la h, \lambda_{\overline{S}_{i-1}}  \ra +
\la h, \lambda_{\overline{B}_{j,i}   }  \ra
=\la h, \lambda_{\overline{S}_{i-1} \sqcup    \overline{B}_{j,i}   }  \ra \leq z(\overline{S}_{i-1} \sqcup    \overline{B}_{j,i}  )
\]

and so

\[
\la h, \lambda_{\overline{B}_{j,i}   }  \ra       \leq    z(\overline{S}_{i-1} \sqcup    \overline{B}_{j,i}  )  -  z(\overline{S}_{i-1})
\]

For $\textcolor{blue}{(\refeqq{eq:55})}$ & $\textcolor{blue}{(\refeqq{eq:66})}$ $\implies$ $\textcolor{blue}{(\refeqq{eq:11})}$, we have

\[
\la h, \lambda_{  \overline{S}_{i-1}\sqcup   \overline{B}_{j,i}       }  \ra
\la h, \lambda_{  S_{1}       }  \ra     +       \dots +   \la h, \lambda_{  S_{i-1}       }  \ra      +\la h, \lambda_{   \overline{B}_{j,i}       }  \ra
\]

\[ \leq          z(\overline{S}_1) + \big (z(\overline{S}_2) -  z(\overline{S}_1)  \big ) + \cdots\cdots  +     \big (z(\overline{S}_{i-1}) -  z(\overline{S}_{i-2})  \big )     +       z(\overline{S}_{i-1} \sqcup    \overline{B}_{j,i}  )  -  z(\overline{S}_{i-1})
\]

\[
z(\overline{S}_{i-1}\sqcup \overline{B}_{j,i})
.\qedhere \]

§.§ Invertible Sheaves From Boolean Functions

We now define a morphism of cospecies

\[
\mathcal{O}: \textsf{BF} \to \text{Mod}
z\mapsto \mathcal{O}_z
\]

whose composition with the external product morphism $\text{Mod}\to \textsf{Mod}$ is a homomorphism of bimonoids $\varphi:\textsf{BF}\to  \textsf{Mod}$. The morphism $\mathcal{O}$ will be Joyal-theoretic, but not strict.

Given a composition $H\in \Sigma[I]$ and a Boolean function $z\in \textsf{BF}[I]$, we have the $\bC[\text{M}_H]$-module $\bC[\text{M}_{H,z}]$ given by

\[
\text{M}_{H,z} := [[H]]_z \cap \bZ I \qquad \text{and} \qquad  \bC[\text{M}_{H,z}]:=  \bigoplus_{h\in  \text{M}_{H,z}} \bC \cdot g^h

We define the invertible sheaf $\mathcal{O}_z$ of $z$ to be the coherent sheaf which, restricted to the affine open $U_H\subset \bC\Sigma^I$, is the coherent sheaf of the $\bC[\text{M}_H]$-module $\bC[\text{M}_{H,z}]$. We now make this explicit.

We make the identification

\[
\mathcal{K}_I=  \text{Frac} \big(   \bC[\text{T}^\vee_I \cap \bZ I]   \big  )
\]

where recall $\mathcal{K}_I$ denotes the field of rational functions on permutohedral space $\bC\Sigma^I$. For each height $a\in \bZ$, we have the $\mathcal{K}_I$-module $\mathcal{K}_{I,a}$ given by

\[
\mathcal{K}_{I,a} :=  \text{Frac} \big(   \bC[\text{T}^\vee_{I,a} \cap \bZ I]      \big  )=  \bC[\text{T}^\vee_{I,a} \cap \bZ I]    \otimes_{ \bC[\text{T}^\vee_I \cap \bZ I]  }   \mathcal{K}_I

We can think of $\mathcal{K}_{I,a}$ as rational functions on permutohedral space which have an ambient degree of divergence equal to $a$. Importantly for the bimonoid structure, this extended notion of rational functions will be closed under pullback along the multiplication $\upmu_{F,G}$.

For $g\in \bC[\text{M}_{H,z}]\subseteq  \bC[\text{T}^\vee_{I,\text{hei}(z)} \cap \bZ I] $ and $f\in  \bC[\text{M}_H]$, we abbreviate

\[
g/f := g\otimes 1/f \in \mathcal{K}_{I,\text{hei}(z)}

For the principal open set $U_f\subseteq U_H$, put

\begin{equation}\label{eq:def}
\mathcal{O}_z (U_f) :=   \big \{ g/f^p  \ \big | \ g\in     \bC[\text{M}_{H,z}], \ p\in \bN   \big \} \subseteq  \mathcal{K}_{I,\text{hei}(z)}
\end{equation}

Assuming this is well-defined, this defines the sheaf $\mathcal{O}_z$ on a basis of $\bC\Sigma^I$, and so defines the sheaf entirely. The stalk of $\mathcal{O}_z$ at $x\in U_H$ is given by

\begin{equation}\label{eq:stalks}
\mathcal{O}_z(x) = \colim_{x\in U} \mathcal{O}_z (U) =     \big \{         g/f\in    \mathcal{K}_{I,\text{hei}(z)} \ \big | \  g\in     \bC[\text{M}_{H,z}], \  f(x)\neq 0     \big \}
\end{equation}

We need to check that (eq:def) is well-defined, and that (eq:stalks) really are the stalks.

We show this by constructing isomorphisms with the subsheaves $\mathcal{O}_w$ of the sheaf of rational functions as follows. Suppose we have principal opens

\[
U_{f_1}\subseteq U_{H_1}
\qquad \text{and} \qquad
U_{f_2}\subseteq U_{H_2}
\qquad \text{with} \qquad
U_{f_1} = U_{f_2} =U

Pick some $d \in \text{T}^\vee_{I,-\text{hei}(z)}\cap \bZ I$. Then we have an isomorphism of vector spaces $\Phi_{\text{hei}(z),d}$ given by

\[
\Phi_{\text{hei}(z),d} : \mathcal{K}_{I,\text{hei}(z)} \xrightarrow{\sim}  \mathcal{K}_{I,\text{hei}(z)-\text{hei}(z)} =  \mathcal{K}_{I,0}=  \mathcal{K}_I
g^h\otimes 1/f\  \mapsto \     g^{h+ d}\otimes 1/f

Let $w=z+z_d$, which is a height zero Boolean function with associated sheaf $\mathcal{O}_w$ described in <ref>. Given $f\in \bC[\text{M}_{H}]$, for the principal open $U_f\subseteq U_H$ we have

\[
\mathcal{O}_w (U_f) =   \big \{ g/f^p \in \mathcal{K}_{I}   \ \big | \ g\in     \bC[\text{M}_{H,w}] , \ p\in \bN   \big \}

See e.g. toric section 4.3. Thus

\[
\Phi_{\text{hei}(z),d} (\mathcal{O}_z (U_{f_1})  )  =   \mathcal{O}_w (U_{f_1}) =   \mathcal{O}_w (U)  =\mathcal{O}_w (U_{f_2}) =  \Phi_{\text{hei}(z),d} (\mathcal{O}_z (U_{f_2})  )

Since $\Phi_{\text{hei}(z),d}$ is an isomorphism, we must have

\[
\mathcal{O}_z (U_{f_1}) =  \mathcal{O}_z (U_{f_2})
\]

as required. More generally, for any $d\in \bZ I$, we have an isomorphism of invertible sheaves

\[
\mathcal{O}_{z\to z+z_{d}} :  \mathcal{O}_z \xrightarrow{\sim} \mathcal{O}_{z+z_d}
g^h\otimes 1/f \  \mapsto  \    g^{h+ d}\otimes 1/f

Finally, we need to define a $2$-cell $\sigma_{I}$ and then check a single coherence law. In particular, we require a natural map of invertible sheaves $\sigma_{z}$ of the form

\[
\sigma_{z} :  \mathcal{O}'_z \to \mathcal{O}_{z'}
\]

for each Boolean function $z\in \textsf{BF}[J]$ and bijection $\sigma:J\to I$. Let $H\in \Sigma[J]$. Over $U_{H'}\subset \bC\Sigma^I$, the pullback $\mathcal{O}'_{z}$ is (naturally isomorphic to) the coherent sheaf of the $\bC[ \text{M}_{ H' } ]$-module

\[
\bC[ \text{M}_{ H,z } ] \otimes     \bC[ \text{M}_{ H' } ]   :=\bC[ \text{M}_{ H,z } ] \otimes_{\bC[ \text{M}_{ H } ]}     \bC[ \text{M}_{ H' } ]

Over $U_{H'}$, $\mathcal{O}_{z'}$ is by definition the coherent sheaf of the $\bC[ \text{M}_{ H' } ]$-module $ \bC[ \text{M}_{ H',z' } ]$. Over $U_{H'}$, we define $\sigma_{z}$ to be the isomorphism of coherent sheaves corresponding to the isomorphism of $\bC[ \text{M}_{ H' } ]$-modules

\[
\bC[ \text{M}_{ H,z } ] \otimes     \bC[ \text{M}_{ H' } ]  \to \bC[ \text{M}_{ H',z' } ]
g\otimes f \mapsto f \cdot  g'

Every element of the left-hand side is of the form $g\otimes 1$, and so this map is also given by $g\otimes 1 \mapsto g'$. We now make $\sigma_{z}$ explicit at the level of stalks. The stalk of the pullback $\mathcal{O}'_z$ at $x\in U_{H'}$ is given by

\[
\mathcal{O}'_z(x) = \mathcal{O}_z(x') \otimes  \mathcal{O}_I(x) \]

\[
=     \big \{         g/f\in    \mathcal{K}_{J,\text{hei}(z)} \, \big | \,  g\in     \bC[\text{M}_{H,z}],\,    f\in     \bC[\text{M}_{H}],  \,  f(x')\neq 0     \big \}
\otimes
\big \{         f_1/f_2\in    \mathcal{K}_{I} \, \big | \,  f_1,f_2\in     \bC[\text{M}_{H'}], \,  f_1(x)\neq 0     \big \}

The stalk of $\mathcal{O}_{z'}$ at $x\in U_{H'}$ is given by

\[
\mathcal{O}_{z'}(x) =   \big \{         g/f\in    \mathcal{K}_{I,\text{hei}(z')} \, \big | \,  g\in     \bC[\text{M}_{H',z'}], \,   f\in     \bC[\text{M}_{H'}],     \,  f(x)\neq 0     \big \}

Then the map $\sigma_{z}$ on the stalk at $x\in U_{H'}$ is given by

\[
g/f \otimes  f_1 / f_2   \,  \mapsto\,  (f_1 \cdot g')/(f_2\cdot f')

This gives a globally well-defined $\sigma_{I}$ as we vary the affine chart $U_{H'}$ by $(\sigma,\mathtt{f})$-functoriality of $\bC\textbf{\textsf{O}}$. We write germs as e.g. $\text{g}=g/f$ and $\text{g}'=g'/f'$. Every element of $  \mathcal{O}'_z(x)$ is of the form $\text{g}\otimes 1$. Then this map may be written as

\[
\text{g}\otimes 1 \mapsto \text{g}'

We have a Joyal-theoretic morphism of cospecies

\[
\mathcal{O} : \textsf{BF} \to \text{Mod}

This is a strong morphism.

Recall $\sigma$-coherence (for cospecies) says

\[
(\sigma \circ \tau)_z  \circ  (\sigma,\tau)_{\mathcal{O}_z} =\underbrace{   \eta_{F''}\big ( (\sigma,\tau)_{z}\big)}_{=\, \text{id}}  \circ \,  \sigma_{z'} \circ \tau'_{z}

At each stalk, the left-hand map is given by

\[
\text{g} \otimes 1 \otimes 1
\mapsto
\text{g} \otimes 1
\mapsto
\text{g}''
\]

and right-hand map is given by

\[
\text{g} \otimes 1 \otimes 1 \mapsto \text{g}' \otimes 1  \mapsto (\text{g}')'= \text{g}''
.\qedhere \]

The restriction of the morphism $\textsf{BF} \to \text{Mod}$ to height zero Boolean functions $\bZ^{[-;2]}\hookrightarrow \textsf{BF}$ recovers the usual construction of invertible sheaves on toric varieties as subsheaves of the sheaf of rational functions. Thus, the essential image of $z\mapsto \mathcal{O}_z$ is all invertible sheaves.

§.§ A Homomorphism of Bimonoids

We have the composition of morphisms $\varphi:= \text{prod} \circ \mathcal{O}$,

\[
\varphi: \textsf{BF} \to \textsf{Mod}
( z_1,\dots, z_k )\mapsto \mathcal{O}_{z_1} \boxtimes \dots \boxtimes \mathcal{O}_{z_k}

We now equip $\varphi$ with the structure of a strong homomorphism of bimonoids, which involves defining $2$-cells ${\mathtt{\Phi}_{F,G}}$ and ${\mathtt{\Psi}_{F,G}}$ and then checking nine coherence laws.

For compositions $G\leq F$, let

\[
F=(S_1,\dots,   S_k)\qquad \text{and} \qquad G=(T_1, \dots,   T_l)

Let $1\leq \mathtt{i}_j \leq k$ be the integer such that $S_{\mathtt{i}_j}$ is the $\mathtt{i}$th lump of $F|_{T_j}$, $1\leq j \leq l$. Let $k(j)$ denote the length of $F|_{T_j}$.

Given an $F$-tuple of Boolean functions $z=(z_1,\dots,z_k)\in \textsf{BF}[F]$, we require a map of $\mathcal{O}_G$-modules ${\mathtt{\Phi}_{F,G}}_{(z_1,\dots,z_k)}$ of the form

\[
{\mathtt{\Phi}_{F,G}}_{(z_1,\dots,z_k)} :
\Updelta^\ast_{F,G}  (  \mathcal{O}_{z_1}  \boxtimes\dots \boxtimes  \mathcal{O}_{z_k}   )
\to
\mathcal{O}_{(z_{1_1} |\dots | z_{k(1)_1}  )} \boxtimes \cdots \boxtimes   \mathcal{O}_{(   z_{1_l} | \dots | z_{k(l)_l}  )}

Let $H=(H_1, \dots, H_l)\in \mathsf{\Sigma}[G]$. Over $U_H\subset \bC\Sigma^G$, the pullback $\Updelta^\ast_{F,G}\mathcal{O}_{(z_1 , \dots, z_k)}=\Updelta^\ast_{F,G}  (  \mathcal{O}_{z_1}  \boxtimes\dots \boxtimes  \mathcal{O}_{z_k}   )$ is (naturally isomorphic to) the coherent sheaf of the $\bC[\text{M}_{H}]$-module

\[
\bigotimes_{1\leq i \leq k}
\Big(
\bC[\text{M}_{H|_{S_i},z_i}] \otimes   \bC[\text{M}_{ (H|_{S_1} ,\dots , H|_{S_k}) }]
\Big)
\otimes
\bC[\text{M}_{H}] \, :=
\]

\[
{\bigotimes_{1\leq i \leq k}}_{\, \bC[\text{M}_{ (H|_{S_1} ,\dots , H|_{S_k}) }]}
\Big(
\bC[\text{M}_{H|_{S_i},z_i}] \otimes_{ \bC[\text{M}_{H|_{S_i}}] }   \bC[\text{M}_{ (H|_{S_1} ,\dots , H|_{S_k}) }]
\Big)
\otimes_{\bC[\text{M}_{ (H|_{S_1} ,\dots , H|_{S_k}) }]}
\bC[\text{M}_{H}]

The external product $\mathcal{O}_{(z_{1_1} |\dots | z_{k(1)_1}  )} \boxtimes \cdots \boxtimes   \mathcal{O}_{(   z_{1_l} | \dots | z_{k(l)_l}  )}$ over $U_H$ is (naturally isomorphic to) the coherent sheaf of the $\bC[\text{M}_{H}]$-module

\[
\bigotimes_{1 \leq j\leq l}  \Big ( \bC[ \text{M}_{H_j,  ( z_{1_j} |\dots | z_{k(j)_j})  } ] \otimes \bC[\text{M}_{H}] \Big )
{\bigotimes_{1 \leq j\leq l}}_{\, \bC[\text{M}_{H}]} \bC[ \text{M}_{H_j,  ( z_{1_j} |\dots | z_{k(j)_j})  } ] \otimes_{\bC[\text{M}_{H_j}]} \bC[\text{M}_{H}]

Over $U_H$, we define ${\mathtt{\Phi}_{F,G}}_{(z_1,\dots,z_k)}$ to be the isomorphism of coherent sheaves which corresponds to the isomorphism of $\bC[\text{M}_{H}]$-modules

\[
\bigotimes_{1\leq i \leq k}
\Big(
\bC[\text{M}_{H|_{S_i},z_i}] \otimes   \bC[\text{M}_{ (H|_{S_1} ,\dots , H|_{S_k}) }]
\Big)
\otimes
\bC[\text{M}_{H}]
\to
\bigotimes_{1 \leq j\leq l}  \Big ( \bC[ \text{M}_{H_j,  ( z_{1_j} |\dots | z_{k(j)_j})  } ] \otimes \bC[\text{M}_{H}]  \Big )
\]

\[
\underbrace{
\bigotimes_{1\leq i \leq k}  ( g^{h_{i}} \otimes 1 ) \otimes 1
\bigotimes_{1\leq j \leq l}  \Big ( \bigotimes_{1_j \leq \mathtt{i}  \leq k(j)_j}  \big( g^{h_{\mathtt{i}_j}}\otimes 1 \big) \Big ) \otimes 1
}_{\text{collect into lumps of $G$}}
\, \mapsto \,
\bigotimes_{1\leq j \leq l} g^{\text{m}_{F|_{T_j}, a_j}  (   \bigtimes_{1_j \leq \mathtt{i}  \leq k(j)_j}  h_{\mathtt{i}_j}   )} \otimes 1
\]

where $a_j= (\text{hei}(z_{1_j} ) , \dots , \text{hei}(z_{k(j)_j} )  )$. Note not every element of the left-hand side is of the form $\bigotimes_{1\leq i \leq k} (g^{h_i}  \otimes 1 ) \otimes 1$, rather the map is given by extending this assignment $\bC[\text{M}_{H}]$-linearly. This map of modules is well-defined by <ref>. It is an isomorphism because the map of <ref> is injective.

We now make ${\mathtt{\Phi}_{F,G}}_{(z_1,\dots,z_k)}$ explicit at the level of stalks. The stalk of $\Updelta^\ast_{F,G}\mathcal{O}_{(z_1,\dots,z_k)}$ at $x\in U_{H}$ is given by

\[
\Updelta^\ast_{F,G}\mathcal{O}_{(z_1,\dots,z_k)}(x) =
\bigotimes_{1\leq i \leq k} \Big ( \mathcal{O}_{z_i}( x|_{S_i} ) \otimes \mathcal{O}_{F} \big (    \Updelta_{F,G}(x) \big   ) \Big ) \otimes \mathcal{O}_G(x)
\]

\[
=\bigotimes_{1\leq i \leq k} \Big ( \big \{         g/f\in    \mathcal{K}_{S_i,\text{hei}(z_i)} \ \big | \  g\in     \bC[\text{M}_{H|_{S_i},z_i}],\,  f(x|_{S_i})\neq 0     \big \} \otimes \mathcal{O}_{F} \big (    \Updelta_{F,G}(x) \big   )  \Big )
\otimes \mathcal{O}_G(x)

The stalk of $\mathcal{O}_{(z_{1_1} |\dots | z_{k(1)_1}  )} \boxtimes \cdots \boxtimes   \mathcal{O}_{(   z_{1_l} | \dots | z_{k(l)_l}  )}$ at $x\in U_{H}$ is given by

\[
\bigotimes_{1\leq j \leq l} \big ( \mathcal{O}_{(z_{1_j} |\dots | z_{k(j)_j}  )}  (x|_{T_j})  \otimes \mathcal{O}_G(x) \big )
\]

\[
=\bigotimes_{1\leq j \leq l} \Big ( \big \{         g/f\in    \mathcal{K}_{T_j,\text{hei}(z_{1_j} |\dots | z_{k(j)_j}  )} \ \big | \  g\in     \bC[\text{M}_{H|_{j},(z_{1_j} |\dots | z_{k(j)_j}  )}],\,  f(x|_{T_j})\neq 0     \big \}
\otimes \mathcal{O}_G(x) \Big )

Then the map of stalks is given by extending the assignment

\[
\bigotimes_{1 \leq i\leq k} (g^{h_i}  \otimes 1 ) \otimes 1
\ \mapsto\
\bigotimes_{1 \leq j\leq l}  \big (g^{ \text{m}_{F|_{T_j}, a_j  }  (   \bigtimes_{1_j \leq \mathtt{i}_j \leq k(j)_j} h_{\mathtt{i}_j}   )} \otimes 1 \big )
\]

$\mathcal{O}_G(x)$-linearly.[ explicitly, $\bigotimes_{i}  (g^{h_i}/f_i \otimes \text{f}_i ) \otimes \text{f}
\	\mapsto \
\bigslant{   (  \prod_{i} \Updelta^\ast_{F,G} \text{f}_i  )\text{f}}{ \prod_{i}  (\Updelta^\ast_{F,G}\pi^\ast_{S_i} f_i  )}
\bigotimes_{j}\big (g^{ \text{m}_{F|_{T_j}, a_j  }  (   \bigtimes_{\mathtt{i}_j}  h_{\mathtt{i}_j}   )} \otimes 1 \big )$] This gives a globally well-defined ${\mathtt{\Phi}_{F,G}}_{(z_1,\dots,z_k)}$ as we vary the affine chart $U_{H}$ by multiplication $\mathtt{f}$-naturality of $\bC\textbf{\textsf{O}}$.  When checking the coherence laws for a homomorphism of bimonoids in <ref>, we shall abbreviate e.g. $h=(g^h \otimes1)\otimes 1$, so that the map gets written as

\begin{equation}\label{eq:Phi}
\bigotimes_{1\leq i \leq k} h_i
\, \mapsto \,
\bigotimes_{1\leq j \leq l} \text{m}_{F|_{T_j}, a_j  } \Big (   \bigtimes_{\mathtt{i}}  h_{\mathtt{i}_j}    \Big )
\end{equation}

Given a $G$-tuple of Boolean functions $z=(z_1, \dots, z_l)\in \textsf{BF}[G]$, we require a map of $\mathcal{O}_F$-modules ${\mathtt{\Psi}_{F,G}}_{( z_1 , \dots , z_l )}$ of the form

\[
{\mathtt{\Psi}_{F,G}}_{( z_1 , \dots , z_l )} :
\upmu^\ast_{F,G} (\mathcal{O}_{  z_1} \boxtimes  \dots \boxtimes    \mathcal{O}_{  z_l}  )
\to
\mathcal{O}_{z_1\talloblong_{S_1}} \boxtimes \dots \boxtimes   \mathcal{O}_{z_l\talloblong_{S_k}}

Let $K=(K_1,\dots, K_k)\in \mathsf{\Sigma}[F]$. Over $U_K\subset \bC\Sigma^F$, the pullback $\upmu^\ast_{F,G} (\mathcal{O}_{  z_1} \boxtimes  \dots \boxtimes    \mathcal{O}_{  z_l}  ) $ is (naturality isomorphic to) the coherent sheaf of the $\bC[\text{M}_K]$-module

\[
\bigotimes_{ 1 \leq j \leq l }
\Big (
\bC[ \text{M}_{K_{1_j} ; \cdots ; K_{k(j)_j}, z_j } ]
\otimes
\bC[\text{M}_{(K_{1_1} ; \cdots ; K_{k(1)_1},  \cdots, K_{1_l}; \cdots ; K_{k(l)_l})}]
\Big )
\otimes   \bC[ \text{M}_{K} ]
\]

\[
{\bigotimes_{ 1 \leq j \leq l }}_{\bC[\text{M}_{(K_{1_1} ; \cdots ; K_{k(1)_1},  \cdots, K_{1_l}; \cdots ; K_{k(l)_l})}]}
\Big (
\bC[ \text{M}_{K_{1_j} ; \cdots ; K_{k(j)_j}, z_j } ]
\otimes_{ \bC[  M_{K_{1_j} ; \cdots ; K_{k(j)_j }}]  }
\bC[\text{M}_{(K_{1_1} ; \cdots ; K_{k(1)_1},  \cdots, K_{1_l}; \cdots ; K_{k(l)_l})}]
\Big )
\]

\[
\otimes_{ \bC[\text{M}_{(K_{1_1} ; \cdots ; K_{k(1)_1},  \cdots, K_{1_l}; \cdots ; K_{k(l)_l})}] }   \bC[ \text{M}_{K} ]

Over $U_K$, the external product $\mathcal{O}_{z_1\talloblong_{S_1}} \boxtimes \dots \boxtimes   \mathcal{O}_{z_l\talloblong_{S_k}}$ is (naturality isomorphic to) the coherent sheaf of the $\bC[\text{M}_{K}]$-module

\[
{\bigotimes_{1\leq i\leq k}}
\Big (\bC[ \text{M}_{ K_i, z\talloblong_{S_i} } ] \otimes  \bC[ \text{M}_{ K  } ]\Big )
{\bigotimes_{1\leq i\leq k}}_{\,  \bC[ \text{M}_{ K } ]}
\Big (\bC[ \text{M}_{ K_i, z\talloblong_{S_i} } ] \otimes_{\bC[ \text{M}_{K_i}]} \bC[ \text{M}_{ K  } ]\Big )

Over $U_K$, we define ${\mathtt{\Psi}_{F,G}}_{( z_1 , \dots , z_l )}$ to be the isomorphism of coherent sheaves which corresponds to the isomorphism of $\bC[\text{M}_{K}]$-modules

\[
\bigotimes_{ 1 \leq j \leq l }
\Big (
\bC[ \text{M}_{K_{1_j} ; \cdots ; K_{k(j)_j}, z_j } ]
\otimes
\bC[\text{M}_{(K_{1_1} ; \cdots ; K_{k(1)_1},  \cdots, K_{1_l}; \cdots ; K_{k(l)_l})}]
\Big )
\otimes   \bC[ \text{M}_{K} ]
\to
{\bigotimes_{1\leq i\leq k}}
\Big (\bC[ \text{M}_{ K_i, z\talloblong_{S_i} } ] \otimes  \bC[ \text{M}_{ K  } ]\Big )
\]

\[
\underbrace{
\bigotimes_{1\leq j\leq l}  ( g^{h_{j}}  \otimes 1)  \otimes 1
\bigotimes_{1\leq j\leq l}  \big(
g^{\text{m}_{	  F|_{T_j}   , a_j }
( \bigtimes_{1_j\leq  \mathtt{i} \leq k(j)_j} d_{\mathtt{i}_j} )} \otimes 1 \big) \otimes 1
}_{\text{factorize each $h_{j}$}}
\  \mapsto \
\bigotimes_{1\leq j\leq l}  \big ( \bigotimes_{1_j\leq  \mathtt{i} \leq k(j)_j}   g^{d_{\mathtt{i}_j}} \otimes 1 \big )
\bigotimes_{1\leq i \leq k} (g^{d_i} \otimes 1)
\]

where $a_j=( \text{hei}(z\talloblong_{S_{1_{j}}}),\dots,  \text{hei}(z\talloblong_{S_{ k(j)_j }} ) )$ and we have assumed each $h_j$ factorizes as

\[
h_{j}= \text{m}_{	  F|_{T_j}   , a_j } \big (\bigtimes_{1_j\leq  \mathtt{i} \leq k(j)_j} d_{\mathtt{i}_j}\big )

i.e. $h_j$ lies in the $ F|_{T_j}$-face of the plate $[[ K_{1_j} ; \dots ; K_{k(j)_j} ]]_{z_j}$. Notice that if the $h_j$ do not factorize in this way, then $\bigotimes_{1\leq j\leq l}  ( g^{h_{j}}  \otimes 1)  \otimes 1=0$. This map of modules is well-defined by <ref>. It is an isomorphism because the map of <ref> is injective.

We now make ${\mathtt{\Psi}_{F,G}}_{( z_1 , \dots , z_l )}$ explicit at the level of stalks. The stalk of $\upmu^\ast_{F,G} (\mathcal{O}_{  z_1} \boxtimes  \dots \boxtimes    \mathcal{O}_{  z_l}  ) $ at $x\in U_K$ is given by

\[
\bigotimes_{1\leq j \leq l}  \Big (  \mathcal{O}_{z_j}\big (  \upmu_{F,G}(x)|_{T_j}\big )      \otimes \mathcal{O}_G\big(  \upmu_{F,G}(x)\big) \Big )        \otimes \mathcal{O}_F(x)
\]

\[
=\bigotimes_{1\leq j \leq l}  \Big ( \big \{         g/f\in    \mathcal{K}_{T_j,\text{hei}(z_j)} \ \big | \  g\in
\bC[\text{M}_{K_{1_j} ; \cdots ; K_{k(j)_j}, z_j}]
,\,  f( \upmu_{F,G}(x)|_{T_j})\neq 0     \big \}       \otimes \mathcal{O}_G\big(  \upmu_{F,G}(x)\big) \Big )        \otimes \mathcal{O}_F(x)

The stalk of $\mathcal{O}_{z_1\talloblong_{S_1}} \boxtimes \dots \boxtimes   \mathcal{O}_{z_l\talloblong_{S_k}}$ at $x\in U_K$ is given by

\[
\bigotimes_{1\leq i \leq k} \big ( \mathcal{O}_{z\talloblong_{S_i}}(x|_{S_i}) \otimes \mathcal{O}_F(x) \big)
\]

\[
=\bigotimes_{1\leq i \leq k} \big (\big \{         g/f\in    \mathcal{K}_{S_i,\text{hei}(z\talloblong_{S_i})} \ \big | \  g\in     \bC[\text{M}_{K_i,z\talloblong_{S_i}}],\,  f(x|_{S_i})\neq 0     \big \}   \otimes \mathcal{O}_F(x) \big)

Then the map of stalks is given by extending the assignment

\[
\bigotimes_{1\leq j\leq l}  ( g^{h_{j}}  \otimes 1)  \otimes 1
\bigotimes_{1\leq j\leq l}  \big(
g^{\text{m}_{	  F|_{T_j}   , a_j }
( \bigtimes_{1_j\leq  \mathtt{i} \leq k(j)_j} d_{\mathtt{i}_j} )} \otimes 1 \big) \otimes 1
\  \mapsto \
\bigotimes_{1\leq j\leq l}  \big ( \bigotimes_{1_j\leq  \mathtt{i} \leq k(j)_j}   g^{d_{\mathtt{i}_j}} \otimes 1 \big )
\bigotimes_{1\leq i \leq k} (g^{d_i} \otimes 1)
\]

$\mathcal{O}_F(x)$-linearly. This gives a globally well-defined ${\mathtt{\Psi}_{F,G}}_{(z_1,\dots,z_k)}$ as we vary the affine chart $U_{K}$ by comultiplication $\mathtt{f}$-naturality of $\bC\textbf{\textsf{O}}$. When checking the coherence laws for a homomorphism of bimonoids in <ref>, we shall abbreviate this map as

\begin{equation} \label{eq:Psi}
\bigotimes_{1\leq j\leq l}  h_{j}    =	\bigotimes_{1\leq j\leq l}  \Big(
\text{m}_{	  F|_{T_j}   , a_j }
\big( \bigtimes_{\mathtt{i}} d_{\mathtt{i}_j}  \big)   \Big)
\,  \mapsto \,
\bigotimes_{1\leq j\leq l}  \bigotimes_{\mathtt{i}}    d_{\mathtt{i}_j}
\end{equation}

The indexing $z\mapsto \mathcal{O}_z$ of invertible sheaves on permutohedral space by Boolean functions, equipped with the $2$-cells ${\mathtt{\Phi}_{F,G}}$ and ${\mathtt{\Psi}_{F,G}}$ as defined above, is a strong homomorphism of bimonoids

\[
\varphi : \textsf{BF} \to \textsf{Mod}

There are nine coherence laws to check. Recall multiplication $\sigma$-naturality coherence says

\[
\underbrace{\varphi_{G'} ( {\mu_{F,G,\sigma}}_z )}_{=\, \text{id}} \, \circ\,  {\sigma}_{\mu_{F,G}(z_1,\dots,z_k)}  \circ  {\mathtt{\Phi}'_{F,G}}_{(z_1,\dots,z_k)}  = {\mathtt{\Phi}_{F',G'}}_{(z_1,\dots,z_k)'} \circ \Updelta^\ast_{F',G'}({\sigma}_{(z_1,\dots,z_k)})
\circ {\mu_{F,G,\sigma}}_{   \mathcal{O}_{(z_1,\dots,z_k)}  }

The left-hand map is given by

\[
\bigotimes_{1\leq i \leq k}   h_{i}
\bigotimes_{1\leq j \leq l}   \bigotimes_{\mathtt{i}}   h_{\mathtt{i}_j}
\mapsto
\bigotimes_{1\leq j \leq l}   \text{m}_{F|_{T_j},a_j} \Big (   \bigtimes_{\mathtt{i}}  h_{\mathtt{i}_j}  \Big  )
\mapsto
\bigotimes_{1\leq j \leq l}  \text{m}_{F'|_{T'_j},a_j} \Big (   \bigtimes_{\mathtt{i}}  h'_{\mathtt{i}_j}  \Big  )
\]

and the right-hand map is given by

\[
{\bigotimes_{1\leq i \leq k}} h_i
\mapsto
{\bigotimes_{1\leq i \leq k}} h_i
\mapsto
\bigotimes_{1\leq i \leq k}  h'_i =  \bigotimes_{1\leq j \leq l}  \bigotimes_{   \mathtt{i}  }  h'_{\mathtt{i}_j}
\mapsto
\bigotimes_{1\leq j \leq l}  \text{m}_{F'|_{T'_j},a_j} \Big (   \bigtimes_{   \mathtt{i}  }  h'_{\mathtt{i}_j}    \Big )

Recall multiplication $\beta$-naturality coherence says

\[
\underbrace{\varphi_{\wt{G}} ( {\mu_{F,G,\beta}}_z )}_{=\, \text{id}}\,  \circ\,  {\beta_{G}}_{\mu_{F,G}( z_1,\dots,z_k )} \circ {\wt{\mathtt{\Phi}_{F,G}}}_{(z_1,\dots,z_k)}
{\mathtt{\Phi}_{\wt{G}F,\wt{G}}}_{\wt{(z_1,\dots,z_k)}}\circ  \Updelta^\ast_{\wt{G}F, \wt{G}}( {\hat{\beta}}_F {\! \,  }_{(z_1,\dots,z_k)}   ) \circ  \mu_{\wt{G}F, \wt{G}}( {\hat{\beta}}_\ta  ) {\circ \mu_{F,G,\beta}}_{\mathcal{O}_z}

The left-hand map is given by[ where the tilde $\wt{F}$ refers to $\hat{\beta}$]

\[
\bigotimes_{1\leq i \leq k}
\mapsto
\bigotimes_{1\leq j \leq l}   \text{m}_{F|_{T_j},a_j} \Big (   \bigtimes_{\mathtt{i} }  h_{\mathtt{i}_j}  \Big  )
\mapsto
\bigotimes_{1 \leq j\leq l}   \text{m}_{\wt{F}|_{T_{\beta(j)}},  a_j  } \Big (   \bigtimes_{ \mathtt{i}  } h_{\hat{\beta}(\mathtt{i}_j)}  \Big  )
\]

and the right-hand map is given by

\[
{\bigotimes_{1\leq i \leq k}}
\mapsto
{\bigotimes_{1\leq i \leq k}}
\mapsto
\bigotimes_{1\leq i \leq k}   h_{\hat{\beta}(i)}
\mapsto
\bigotimes_{1 \leq j\leq l}   \text{m}_{\wt{F}|_{T_{\beta(j)}},  a_j  } \Big (   \bigtimes_{ \mathtt{i}  } h_{\hat{\beta}(\mathtt{i}_j)}  \Big )

Comultiplication $\sigma$-naturality and $\beta$-naturality coherence follow similarly. Recall associativity coherence says

\[
\underbrace{\varphi_E ({\mu_{F,G,E}}_{(z_1,\dots, z_k)})}_{=\, \text{id}} \circ  {\mathtt{\Phi}_{G,E}}_{\mu_{F,G}(z_1,\dots, z_k)}  \circ  \Updelta^\ast_{G,E} (   {\mathtt{\Phi}_{F,G}}_{(z_1,\dots, z_k)}  )
{\mathtt{\Phi}_{F,E}}_{(z_1,\dots, z_k)} \circ {\mu_{F,G,E}}_{\mathcal{O}_{(z_1,\dots, z_k)}}

\[
F=(S_1,\dots, S_i,\dots , S_k), \qquad G=(T_1,\dots, T_j,\dots, T_l),  \qquad E=(U_1, \dots, U_r, \dots, U_e)

Let $\mathtt{i}_j$ be as before. Let $1\leq \mathtt{j}_r \leq l$ be the integer such that $T_{\mathtt{j}_r}$ is the $\mathtt{j}$th lump of $G|_{U_r}$, and let $1\leq {\mathtt{i}_\mathtt{j}}_r\leq k$ such that $S_{{\mathtt{i}_\mathtt{j}}_r}$ is the $\mathtt{i}$th lump of $F|_{T_{\mathtt{j}_r}}$. Then the left-hand map is given by

\[
\bigotimes_{1\leq i \leq k} h_{i}
\bigotimes_{1\leq j \leq l}  \bigotimes_{\mathtt{i}} h_{\mathtt{i}_j}
\mapsto
\bigotimes_{1\leq j \leq l} \text{m}_{F|_{T_j},a_j} \Big (   \bigtimes_{\mathtt{i}}  h_{\mathtt{i}_j}  \Big  )
\bigotimes_{1\leq r \leq e}  \bigotimes_{{\mathtt{j}}}  \text{m}_{F|_{T_{\mathtt{j}_r}}, a_j } \Big (   \bigtimes_{{\mathtt{i}}}  h_{ {\mathtt{i}_\mathtt{j}}_{r}   }  \Big  )
\]

\[
\mapsto
\bigotimes_{1\leq r \leq e}  \text{m}_{G|_{U_r}, a_r}   \bigg ( \bigtimes_{{\mathtt{j}}}  \text{m}_{F|_{T_{\mathtt{j}_r}}, a_{\mathtt{j}_r} } \Big (   \bigtimes_{{\mathtt{i}}}  h_{ {\mathtt{i}_\mathtt{j}}_{r}   }  \Big  )  \bigg)
\]

and the right-hand map is given by

\[
\bigotimes_{1\leq i \leq k} h_{i} \mapsto 	\bigotimes_{1\leq i \leq k} h_{i} \mapsto
\bigotimes_{1\leq r \leq e}  \text{m}_{F|_{U_r}, a_r} \Big (  \bigtimes_{\mathtt{i}}  h_{ \mathtt{i}_{r}   }     \Big)

However, we have

\[
\bigotimes_{1\leq r \leq e}  \text{m}_{G|_{U_r},a_r}   \bigg ( \bigtimes_{{\mathtt{j}}}  \text{m}_{F|_{T_{\mathtt{j}_r}},a_{\mathtt{j}_r}} \Big (   \bigtimes_{{\mathtt{i}}}  h_{ {\mathtt{i}_\mathtt{j}}_{r}   }  \Big  )  \bigg)
\bigotimes_{1\leq r \leq e}  \text{m}_{F|_{U_r}, a_r} \Big (  \bigtimes_{\mathtt{i}}  h_{ \mathtt{i}_{r}   }     \Big)
\]

directly from the definition of $\text{m}_{F,(a_1,\dots, a_k)}$ (the exact property we are using here is actually the associativity of the monoid mentioned in <ref>). Coassociativity coherence says

\[
\underbrace{\varphi_F ({\Delta_{F,G,E}}_{(z_1,\dots, z_{e})})}_{=\, \text{id}} \circ {\mathtt{\Psi}_{F,G}}_{\Delta_{G,E}(z_1,\dots, z_{e})}  \circ  \upmu^\ast_{F,G} (   {\mathtt{\Psi}_{G,E}}_{(z_1,\dots, z_{e})}  )
{\mathtt{\Psi}_{F,E}}_{(z_1,\dots, z_{e})} \circ {\mu_{F,G,E}}_{\mathcal{O}_{(z_1,\dots, z_{e})}}

The left-hand map is given by

\[
\bigotimes_{1\leq r\leq {e}}    h_{r}
\bigotimes_{1\leq r\leq {e}}    \text{m}_{G|_{U_r},a_r} \Big( \bigtimes_{\mathtt{j}} d_{\mathtt{j}_r}  \Big)
\  \mapsto \
\bigotimes_{1\leq r\leq e}  \bigotimes_{\mathtt{j}}    d_{\mathtt{j}_r}
\bigotimes_{1\leq r\leq e}   \bigotimes_{\mathtt{j}}     \text{m}_{ F|_{T_{\mathtt{j}_r}}, a_{\mathtt{j}_r} } \Big (  \bigtimes_\mathtt{i}   c_{{\mathtt{i}_\mathtt{j}}_r} \Big )
\]

\[
\mapsto
\bigotimes_{1\leq r\leq e}   \bigotimes_{\mathtt{j}}     \bigotimes_{{\mathtt{i}}}   c_{{\mathtt{i}_\mathtt{j}}_r}  =\bigotimes_{1\leq r\leq e}   \bigotimes_{\mathtt{i}}    c_{\mathtt{i}_r}
\]

and the right-hand map is given by

\[
\bigotimes_{1\leq r\leq e}  h_{r}    =	\bigotimes_{1\leq r\leq e}  \bigg(
\text{m}_{	  F|_{U_r}   , a_r }
\Big( \bigtimes_{\mathtt{i}} c_{\mathtt{i}_r}  \Big)   \bigg)
\,  \mapsto \,
\bigotimes_{1\leq r\leq e}   \bigotimes_{\mathtt{i}}    c_{\mathtt{i}_r}
\]

(where we have again used the associativity of the monoid mentioned in <ref>). Recall the bimonoid coherence law is

\[
\varphi_G({\text{B}_{F,G,A}}_{\ta})
\circ
\circ
{\beta_{FG}}_{ \Delta_{FG,F} (z)} \circ
\Updelta^\ast_{GF,G}(\wt{{\mathtt{\Psi}_{FG,F}}_z}) \]

\[
\circ
\circ
\upmu^\ast_{G,A}( {\mathtt{\Phi}_{F,A}}_{z}  ) \circ {\text{B}_{F,G,A}}_{\varphi_F(\ta)}

\[
F=(S_1,\dots, S_i, \dots, S_k), \quad G=(T_1,\dots, T_j, \dots, T_l),\]

\[
FG=(X_1,\dots, X_n,\dots, X_L),\quad GF=(Y_1,\dots,Y_n,\dots , Y_L), \quad  A=(A_1,\dots,A_m,\dots , A_{a})

Let $1\leq \mathtt{i}_m \leq k$ be the integer such that $S_{\mathtt{i}_m}$ is the $\mathtt{i}$th lump of $F|_{A_m}$, let $1\leq \mathtt{j}_m \leq j$ such that $T_{\mathtt{j}_m}$ is the $\mathtt{j}$th lump of $G|_{A_m}$, let $1\leq \mathtt{n}_i \leq L$ such that $X_{\mathtt{n}_i}$ is the $\mathtt{n}$th lump of $FG|_{S_i}$, and let $1\leq \mathtt{n}_j \leq L$ such that $Y_{\mathtt{n}_j}$ is the $\mathtt{n}$th lump of $GF|_{T_j}$. Let $1\leq \mathtt{n}_{\mathtt{i}_m}\leq L$ such that $X_{\mathtt{n}_{\mathtt{i}_m}}$ is the $\mathtt{n}$th lump of $FG|_{S_{\mathtt{i}_m}}$, and let $1\leq \mathtt{n}_{\mathtt{j}_m}\leq L$ such that $Y_{\mathtt{n}_{\mathtt{j}_m}}$ is the $\mathtt{n}$th lump of $GF|_{T_{\mathtt{j}_m}}$.

Then the left-hand map is given by

\[
\bigotimes_{1\leq i\leq k} h_i = \bigotimes_{1\leq i\leq k}  \text{m}_{  FG|_{S_i} , a_i } \Big ( \bigtimes_{\mathtt{n}}   d_{\mathtt{n}_i} \Big )
\  \mapsto \
\bigotimes_{1\leq i\leq k}  \bigotimes_{\mathtt{n}}  d_{ \mathtt{n}_i  }
\]

\[
\mapsto\
\underbrace{\bigotimes_{1\leq i\leq k}  \bigotimes_{\mathtt{n}}     d_{\beta(\mathtt{n}_i)} = 	\bigotimes_{1\leq j\leq l}  \bigotimes_{\mathtt{n}}  e_{\mathtt{n}_j}}_{\text{where $e_n:= d_{ \beta( n )}$}}
\ \mapsto \
\bigotimes_{1\leq j \leq l}   \text{m}_{ GF|_{T_j} , a_j } \Big (  \bigtimes_{\mathtt{n}} e_{ \mathtt{n}_j }   \Big   )
\]

and the right-hand map is given by

\[
\bigotimes_{1\leq i\leq k} h_i
\bigotimes_{1\leq m\leq a}   \bigotimes_{\mathtt{i}}
\]

\[
\  \mapsto \
\bigotimes_{1\leq m\leq  a}  \text{m}_{ F|_{A_m} ,a_m} \Big  (  \bigtimes_{\mathtt{i}}  h_{\mathtt{i}_m} \Big  )
\bigotimes_{1\leq m\leq  a}  \text{m}_{ F|_{A_m} ,a_m}  \bigg ( \bigtimes_{\mathtt{i}} \text{m}_{FG|_{S_{\mathtt{i}_m}}, a_{\mathtt{i}_m}}\Big( \bigtimes_{\mathtt{n}}  d_{\mathtt{n}_{\mathtt{i}_m}} \Big  )      \bigg )
\]

\[
= \bigotimes_{1\leq m\leq  a}  \text{m}_{ F|_{A_m} ,a_m} \bigg ( \bigtimes_{\mathtt{j}} \text{m}_{GF|_{T_{\mathtt{j}_m}} , a_{\mathtt{j}_m} }\Big( \bigtimes_{\mathtt{n}}  e_{\mathtt{n}_{\mathtt{j}_m}} \Big  )      \bigg )
\ \mapsto \
\bigotimes_{1\leq m\leq  a}   \bigotimes_{\mathtt{j}} \text{m}_{GF|_{T_{\mathtt{j}_m}} , a_{\mathtt{j}_m} } \Big( \bigtimes_{\mathtt{n}}  e_{\mathtt{n}_{\mathtt{j}_m}} \Big  )
\]

\[
\bigotimes_{1\leq j \leq l}   \text{m}_{ GF|_{T_j} ,a_j} \Big (  \bigtimes_{\mathtt{n}} e_{ \mathtt{n}_j }   \Big   )
.\qedhere \]

The restriction of the homomorphism $\varphi : \textsf{BF} \to \textsf{Mod}$ to the bimonoid generalized permutohedra $\textsf{GP}\hookrightarrow \textsf{BF}$, as defined in <cit.>, is an indexing of the nef invertible sheaves, as a consequence of e.g. <cit.>.

§.§ The Bialgebra of Global Sections of Invertible Sheaves

Let $\bC\textbf{\textsf{BF}}$ be the $\textsf{BF}$-graded bialgebra which is the pullback of the $\textsf{Mod}$-graded bialgebra $\bC\textbf{\textsf{Mod}}$ along the homomorphism $\varphi: \textsf{BF} \to \textsf{Mod}$, as constructed in <ref>. Thus

\[
\bC\textbf{\textsf{BF}}_{(z_1,\dots, z_k)}[F] =    \Gamma_{\bC\Sigma^{F}}(\mathcal{O}_{z_1} \boxtimes \dots \boxtimes  \mathcal{O}_{z_k})

Let $\bC\textbf{BF}$ be the Joyalization of $\bC\textbf{\textsf{BF}}$, thus

\[
\bC\textbf{BF}_{(z_1,\dots, z_k)}[F] :=    \Gamma_{\bC\Sigma^{S_1}}(\mathcal{O}_{z_1}) \otimes  \dots \otimes   \Gamma_{\bC\Sigma^{S_k}}(\mathcal{O}_{z_k})

We now show that $\bC\textbf{\textsf{BF}}$ is a strongly Joyal-theoretic bialgebra. The external product of global sections defines a morphism of $\textsf{BF}$-graded vector cospecies $ \bC\textbf{BF}   \to \bC\textbf{\textsf{BF}}$ over the identity, given by

\[
\text{prod}: \bC\textbf{BF}   \to \bC\textbf{\textsf{BF}},\qquad    s_1 \otimes  \dots \otimes s_k\, \mapsto\,   s_{1} \boxtimes \dots \boxtimes  s_{k} := \pi_{S_1}^\ast s_1 \otimes  \dots \otimes  \pi_{S_k}^\ast s_k

The external product of global sections

\[
\text{prod}: \bC\textbf{BF}   \to \bC\textbf{\textsf{BF}}
s_{1} \otimes \dots \otimes  s_{k}   \mapsto   s_{1} \boxtimes \dots \boxtimes  s_{k}
\]

is an isomorphism of $\textsf{BF}$-graded vector cospecies.

We have

\[
\Gamma_{\bC\Sigma^{S_1}}(\mathcal{O}_{z_1}) \otimes  \dots \otimes   \Gamma_{\bC\Sigma^{S_k}}(\mathcal{O}_{z_k})
= \bC\Big [ \bigcap_{H_1\in \Sigma[S_1]}   \text{M}_{H_1, z_1} \Big ] \otimes \dots \otimes   \bC\Big [ \bigcap_{H_k\in \Sigma[S_k]}   \text{M}_{H_k, z_k} \Big ]

Consider an affine open of the form $U_H=U_{H_1}\times \dots \times U_{H_k}$ of $\bC\Sigma^{F}$. Tensoring coherent sheaves corresponds to tensoring their modules on affine charts, and so

\[
\Gamma_{U_H}(\mathcal{O}_{z_1} \boxtimes \dots \boxtimes  \mathcal{O}_{z_k})
(\bC[ \text{M}_{H_1,z_1} ]  \otimes \bC[ \text{M}_{H} ] )\otimes  \dots \otimes  (\bC[ \text{M}_{H_k,z_k} ]\otimes \bC[ \text{M}_{H} )

We make the identification

\[
(\bC[ \text{M}_{H_1,z_1} ]  \otimes \bC[ \text{M}_{H} ] )\otimes  \dots \otimes  (\bC[ \text{M}_{H_k,z_k} ]\otimes \bC[ \text{M}_{H} )
\bC[ \text{M}_{H_1,z_1}   \times \dots \times \text{M}_{H_k,z_k} ]

\[
\Gamma_{\bC\Sigma^{F}}(\mathcal{O}_{z_1} \boxtimes \dots \boxtimes  \mathcal{O}_{z_k}) = \bigcap_H \bC\Big[  \text{M}_{H_1,z_1}   \times \dots \times  \text{M}_{H_k,z_k} \Big ]
= \bC\Big[ \bigcap_H \big ( \text{M}_{H_1,z_1}   \times \dots \times  \text{M}_{H_k,z_k}\big ) \Big ]

Then the external product of global sections is given by the isomorphism of vector spaces

\[
\bC\Big [ \bigcap_{H_1\in \Sigma[S_1]}   \text{M}_{H_1, z_1} \Big ] \otimes \dots \otimes   \bC\Big [ \bigcap_{H_k\in \Sigma[S_k]}   \text{M}_{H_k, z_k} \Big ]
\to
\bC\Big[ \bigcap_H \big ( \text{M}_{H_1,z_1}   \times \dots \times  \text{M}_{H_k,z_k}\big ) \Big ]
\]

\[
g^{h_1} \otimes \dots \otimes g^{h_k}  \mapsto  g^{(h_1,\dots,h_k)}.\qedhere
\]

The $\textsf{BF}$-graded bialgebra $\bC\textbf{\textsf{BF}}$ is strongly Joyal-theoretic.

For the multiplication, we need to show factorization

\[
\Updelta^\ast_{F,G}\Big ( \bigotimes_{1\leq i \leq k} s_i  \Big )  =  \bigotimes_{1\leq j \leq l} \Updelta^\ast_{F|_{T_j}}\Big(  \bigotimes_\mathtt{i}  s_{\mathtt{i}_j}\Big )

We have that the multiplication pulled-back to $\bC\textbf{\text{BF}}$ along $\text{prod}$ is given by

\[
\Updelta^\ast_{F,G} :  \bigotimes_{1\leq i \leq k} \Gamma_{\bC\Sigma^{S_i}}(\mathcal{O}_{z_i})
\to
\bigotimes_{1\leq j \leq l}    \Gamma_{\bC\Sigma^{T_j}}(\mathcal{O}_{  (  z_{1_j}  | \dots | z_{k(j)_j} )  })
\]

\[
\bigotimes_{1\leq i \leq k}  s^{h_i}
\mapsto
\bigotimes_{1\leq j \leq l}  s^{\text{m}_{F|_{T_j}, a_j  } \big (   \bigtimes_{\mathtt{i}_j}  h_{\mathtt{i}_j}    \big )}
\]

and indeed we have factorization because

\[
\bigotimes_{1\leq j \leq l}  s^{\text{m}_{F|_{T_j}, a_j  } \big (   \bigtimes_{\mathtt{i}_j}  h_{\mathtt{i}_j}    \big )}
\Updelta^\ast_{F|_{T_j}}\Big(  \bigotimes_\mathtt{i}  s^{h_{\mathtt{i}_j}} \Big )

For the comultiplication, we need to show factorization

\[
\upmu^\ast_{F,G}\Big ( \bigotimes_{1\leq j \leq l} s_j  \Big )  =  \bigotimes_{1\leq j \leq l} \upmu^\ast_{F|_{T_j}}(  s_j  )

We have that the comultiplication pulled-back to $\bC\textbf{\text{BF}}$ along $\text{prod}$ is given by

\[
\bigotimes_{1\leq j\leq l}      s^{h_{j}}
\bigotimes_{1\leq j\leq l}     s^{\text{m}_{	  F|_{T_j}   , a_j}  ( \bigtimes_{\mathtt{i}_j} h_{\mathtt{i}_j}  )}
\,  \mapsto \,
\bigotimes_{1\leq j\leq l}  \bigotimes_{\mathtt{i}}    s^{h_{\mathtt{i}_j}}
\]

and indeed we have factorization because

\[
\bigotimes_{\mathtt{i}}    s^{h_{\mathtt{i}_j}}   =\upmu^\ast_{F|_{T_j}}(  s^{h_j}  )
.\qedhere \]

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Doc. Math., 13:207–273, 2008.

[28]
Konrad Schultka.
Toric geometry and regularization of Feynman integrals.
arXiv preprint arXiv:1806.01086, 2018.

[29]
Jacques Tits.
Buildings of spherical type and finite BN-pairs.
Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin-New
York, 1974.

Thus $H= (  S_1,\dots, S_{k_S} ,T_1,\dots,T_{k_T})$. Recall

\[
[[ H ]]_z   =  \bigcap_{(A,B)\leq H} \big \{     h\in  \text{T}^\vee_{I,z}   :       \la h, \lambda_{A}  \ra \leq z(A)   \big  \}

\[
[[ K_i    ]]_{z\talloblong_{S_i}}
=  \bigcap_{(A,B)\leq K_i } \big \{     h\in  \text{T}^\vee_{S_i,z\talloblong_{S_i}}   :       \la h, \lambda_{A}  \ra \leq z\! \talloblong_{S_i}(A)   \big  \}

This is the region of $\text{T}^\vee_{S,z\talloblong_S}$ given by the inequalities

\[
\la h, \lambda_{\overline{S}_i }  \ra \leq z(\overline{S}_i)
\]

for $1\leq j \leq k_S$. Similarly, we have

\[
[[ H \! \!  \fatslash_{\, T}   ]]_{z \! \!  \fatslash_{\, T}}
=  \bigcap_{(A,B)\leq H \! \!  \fatslash_{\, T} } \big \{     h\in  \text{T}^\vee_{T,z \! \!  \fatslash_{\, T}}   :       \la h, \lambda_{A}  \ra \leq z \! \! \!  \fatslash_{\, T}(A)   \big  \}

This is the region of $\text{T}^\vee_{T,z \! \!  \fatslash_{\, T}}$ given by the inequalities

\[
\la h, \lambda_{  T_1 \sqcup\dots \sqcup  T_j }  \ra \leq z(S \sqcup T_1 \sqcup\dots \sqcup  T_j)  -z(S)
\]

for $1\leq j \leq k_T$.

We first prove

\[
\Delta_{G,A} \circ \mu_{ F,A } =   \mu_{GF,G} \circ \beta \circ  \Delta_{FG,F }
\]

when $l(F)-l(A)=1$ and $l(G)-l(A)=1$. Indeed

\[
\Delta_{G,A} \circ \mu_{ F,A } =
(\text{id} \otimes \dots \otimes \text{id} \otimes \Delta_{X,Y}\otimes \text{id} \otimes \dots \otimes \text{id})
\circ
(\text{id} \otimes \dots \otimes \text{id} \otimes \mu_{S,T}\otimes \text{id} \otimes \dots \otimes \text{id})
\]

\[
= \text{id} \otimes \dots \otimes \text{id} \otimes (\Delta_{X,Y}\circ  \mu_{S,T}) \otimes \text{id} \otimes \dots \otimes \text{id}
\]

\[
= \text{id} \otimes \dots \otimes \text{id} \otimes \underbrace{(\mu_{S\cap X,T\cap X}\otimes \mu_{S\cap Y,T\cap Y}) \circ B \circ (\Delta_{S\cap X,S\cap Y}\otimes \Delta_{T\cap X, T \cap Y})}_{\text{Joyal bimonoid}} \otimes \text{id} \otimes \dots \otimes \text{id}
\]

\[
=\mu_{GF,G} \circ \beta \circ  \Delta_{FG,F }

We have

\[
\Delta_{X,Y} \circ \mu_{ST,U} \circ  (\mu_{S,T} \otimes \text{id})
\]

\[
\underbrace{   (\mu_{ST\cap X,U\cap X}\otimes \mu_{ST\cap Y,U\cap Y}) \circ \beta \circ (\Delta_{ST\cap X, ST\cap Y}\otimes \Delta_{U\cap X,U\cap Y})  }_{\text{Joyal bimonoid}}   \circ (\mu_{S,T} \otimes \text{id})
\]

\[
(\mu_{ST\cap X,U\cap X}\otimes \mu_{ST\cap Y,U\cap Y}) \circ \beta \circ  (\underbrace{\mu_{S,T} \circ \beta \circ \Delta    }_{\text{Joyal bimonoid}}     \otimes \Delta_{U\cap X,U\cap Y}  )
\]

\[
\Delta_{G,A} \circ \mu_{F,A}
\Delta_{G_{1},G_2}   \circ \dots \circ    \Delta_{G_{L-1},G_L}
\circ
\mu_{F_{L-1},F_L}  \circ \dots \circ  \mu_{F_{1},F_2}
\]

\[
\Delta_{G,A} \circ \mu_{F,A}
\Delta_{G,A}
\circ
\mu_{F_{2},A}   \circ  \mu_{F,F_2}
\]

We also require a map of modules $\beta_{z}$ of the form

\[
\beta_{z} :  \wt{\mathcal{O}}_z \to \mathcal{O}_{\tilde{z}}

Over $U_{\wt{H}}$, the pullback $\wt{\mathcal{O}}_{z}$ is (naturally isomorphic to) the coherent sheaf of the $\bC[ \text{M}_{ \wt{H} } ]$-module

\[
\bC[ \text{M}_{ H,z } ] \otimes   \bC[ \text{M}_{ \wt{H} } ]:= \bC[ \text{M}_{ H,z } ] \otimes_{\bC[ \text{M}_{ H } ]}     \bC[ \text{M}_{ \wt{H} } ]

The stalk at $\tilde{x}\in U_{\wt{H}}$ is given by

\[
\wt{\mathcal{O}}_z(\tilde{x}) = \mathcal{O}_z(x)=     \big \{         g/f\in    \mathcal{K}_{I,\text{hei}(z)} \ \big | \  g\in     \bC[\text{M}_{H,z}], \  f(x)\neq 0     \big \}
\]

with the topology defined as above for $\mathcal{O}'_{z}$, but localizing the module $\bC[ \text{M}_{ H,z } ] \otimes   \bC[ \text{M}_{ \wt{H} } ]$ instead. The stalk of $\mathcal{O}_{\tilde{z}}$ at $\tilde{x}\in U_{\wt{H}}$ is given by

\[
\mathcal{O}_{\tilde{z}}(\tilde{x}) =   \big \{         \tilde{g}/\tilde{f}\in    \mathcal{K}_{I,\text{hei}(\tilde{z})} \ \big | \  \tilde{g}\in     \bC[\text{M}_{\wt{H},\tilde{z}}], \  \tilde{f}(\tilde{x})\neq 0     \big \}

Then the map $\beta_{z}$ is given by

\[
\beta_{z}|_{\tilde{x}}  :    \wt{\mathcal{O}}_z( x ) \to   \mathcal{O}_{\tilde{z}}( x )
g/f \otimes 1 \mapsto \tilde{g}/\tilde{f}

This is well-defined by $(\beta,\mathtt{f})$-functoriality of $\bC\textbf{\textsf{O}}$.

We need to check the three coherence laws for morphisms of cospecies. Recall $(\sigma,\beta)$-coherence says

\[
\underbrace{\eta_{\wt{F}'} \big ( (\sigma,\beta)_{z} \big )}_{=\, \text{id}} \circ  \beta_{z'}   \circ \wt{\sigma}_z
{\sigma}_{\tilde{z}}  \circ \beta'_z  \circ  (\sigma,\beta)_{\mathcal{O}_z}
\]

At each stalk, the left-hand map is given by

\[
\text{g} \otimes 1  \otimes 1 \mapsto   \text{g}' \otimes 1 \mapsto    \wt{(\text{g}')}  =    (\tilde{\text{g}})'
\]

and right-hand map is given by

\[
\text{g} \otimes 1  \otimes 1    \mapsto     \text{g} \otimes 1  \otimes 1   \mapsto  \tilde{\text{g}} \otimes 1 \mapsto    (\tilde{\text{g}})'

\[
\text{p}[\emptyset] :=  1_\textsf{C}
\text{p}[\text{id}_{\emptyset}] :=  \text{id}_{1_\textsf{C}},
\]

\[
\text{p}[I] :=  \textsf{p}[I]
\text{p}[\sigma] := \textsf{p}[\sigma]
\]

for $I\neq \emptyset$. When $\textsf{C}$ is a $2$-category, we also need to put

\[
\text{p}[ \sigma,\tau ] :=   \text{p}\big [ \sigma,\tau, (I)\big ]

We have the functor

\[
\text{Joy}:    \textsf{BraidSp} \to \textsf{JoyalSp}_+
\textsf{p} \mapsto \text{Joy}( \textsf{p} )
\eta \mapsto \text{Joy}(\eta)
\]

where $\text{Joy}( \textsf{p} )=\text{p}$ as constructed above, and

\[
\text{Joy}(\eta)_{\emptyset}:=\text{id}_{1_\textsf{C}}
\text{Joy}(\eta)_{I}:=\eta_{(I)}

When $\textsf{C}$ is a $2$-category, we also need to put

\[
\text{Joy}(\eta)_{\sigma, I}:=   \eta_{\sigma, (I)}

We have the endofunctor

\[
\text{F}:\textsf{BraidSp} \to \textsf{BraidSp}
\text{F}:= \textsf{Br} \circ \text{Joy}

Now the second $\mu$... after the first permutation

\[\begin{tikzcd}[row sep=0.5cm, column sep=0.2cm]
{T_1} && {T_2} && {T_3} && {T_4} \\
&&&&& \mu & \mu \\
&&& {} & \mu & \Delta & \Delta \\
&&& \mu & \Delta & \Delta \\
&& \mu & \mu & \Delta \\
&& \Delta & \Delta \\
&& {S_1} && {S_2} & {S_3}
\arrow[no head, from=5-4, to=6-4]
\arrow[no head, from=4-4, to=1-1]
\arrow[no head, from=3-5, to=1-3]
\arrow[no head, from=2-6, to=1-5]
\arrow[no head, from=6-4, to=7-5]
\arrow[no head, from=4-4, to=5-5]
\arrow[no head, from=4-5, to=5-4]
\arrow[no head, from=4-5, to=3-5]
\arrow[from=5-5, to=7-6]
\arrow[no head, from=3-5, to=4-6]
\arrow[no head, from=3-6, to=4-5]
\arrow[no head, from=4-6, to=5-5]
\arrow[no head, from=2-7, to=3-7]
\arrow[no head, from=3-6, to=2-6]
\arrow[no head, from=2-6, to=3-7]
\arrow[no head, from=2-7, to=3-6]
\arrow[no head, from=4-6, to=3-7]
\arrow[no head, from=2-7, to=1-7]
\arrow[no head, from=5-3, to=6-3]
\arrow[no head, from=5-3, to=6-4]
\arrow[no head, from=5-4, to=6-3]
\arrow[no head, from=6-3, to=7-3]
\arrow[no head, from=5-3, to=4-4]
\end{tikzcd}\]

After fully permuting

\[\begin{tikzcd}[row sep=0.2cm, column sep=0.2cm]
& {T_1} & {T_2} & {T_3} && {T_4} \\
&&&& \mu & \mu \\
& \mu & \mu & \mu & \mu \\
\mu && \mu & \Delta & \Delta & \Delta \\
&& \Delta & \Delta & \Delta \\
& \Delta & \Delta & \Delta \\
& {S_1} & {S_2} && {S_3}
\arrow[no head, from=3-2, to=1-2]
\arrow[no head, from=3-3, to=1-3]
\arrow[no head, from=2-5, to=1-4]
\arrow[no head, from=6-3, to=7-3]
\arrow[no head, from=3-2, to=6-4]
\arrow[no head, from=4-4, to=5-3]
\arrow[no head, from=6-4, to=7-5]
\arrow[no head, from=3-3, to=5-5]
\arrow[no head, from=3-5, to=4-4]
\arrow[no head, from=5-5, to=6-4]
\arrow[no head, from=2-6, to=4-6]
\arrow[no head, from=2-5, to=4-6]
\arrow[no head, from=2-6, to=3-5]
\arrow[no head, from=5-5, to=4-6]
\arrow[no head, from=2-6, to=1-6]
\arrow[no head, from=4-1, to=6-2]
\arrow[no head, from=4-1, to=6-3]
\arrow[no head, from=5-3, to=6-2]
\arrow[no head, from=6-2, to=7-2]
\arrow[no head, from=4-1, to=3-2]
\arrow[no head, from=4-3, to=5-4]
\arrow[no head, from=4-3, to=5-3]
\arrow[no head, from=5-4, to=6-3]
\arrow[no head, from=4-3, to=3-3]
\arrow[no head, from=3-4, to=4-5]
\arrow[no head, from=3-5, to=4-5]
\arrow[no head, from=3-4, to=4-4]
\arrow[no head, from=4-5, to=5-4]
\arrow[no head, from=3-4, to=2-5]
\end{tikzcd}\]

That is, we decompose the higher (co)multiplication into the composition of several binary (co)multiplications, since this is all that the Joyal bimonoid axiom uses. We can now substitute in several instances of the Joyal bimonoid axiom on the left-hand side, e.g.

\[\begin{tikzcd}[column sep=tiny,row sep=tiny]
& {T_1} && {T_2} &&&&& {T_1} && {T_2} &&&&&& {T_1} && {T_2} \\
&& I \\
&& I &&& \mapsto && {S_1 S_2} &&& {S_3} &&& \mapsto && {T_1} & {T_2} && {S_3} \\
& {S_1 S_2} &&&&&& {S_1 S_2} &&&&&&&& {S_1} & {S_2} \\
{S_1} && {S_2} && {S_3} && {S_1} && {S_2} &&&& {S_3} & {} & {S_1} &&& {S_2} &&& {S_3}
\arrow[squiggly, no head, from=3-3, to=2-3]
\arrow[no head, from=2-3, to=1-2]
\arrow[no head, from=2-3, to=1-4]
\arrow[no head, from=1-11, to=3-11]
\arrow[no head, from=1-9, to=3-11]
\arrow[no head, from=5-7, to=4-8]
\arrow[no head, from=5-9, to=4-8]
\arrow[squiggly, no head, from=4-8, to=3-8]
\arrow[no head, from=3-8, to=1-11]
\arrow[no head, from=1-9, to=3-8]
\arrow[no head, from=1-17, to=3-19]
\arrow[no head, from=1-19, to=3-19]
\arrow[no head, from=1-17, to=3-16]
\arrow[no head, from=1-19, to=3-17]
\arrow[no head, from=3-16, to=4-16]
\arrow[no head, from=3-17, to=4-17]
\arrow[no head, from=3-17, to=4-16]
\arrow[no head, from=3-16, to=4-17]
\arrow[no head, from=4-16, to=5-15]
\arrow[no head, from=4-17, to=5-18]
\arrow[no head, from=3-19, to=5-21]
\arrow[from=3-3, to=5-5]
\arrow[from=3-3, to=4-2]
\arrow[from=4-2, to=5-1]
\arrow[from=4-2, to=5-3]
\arrow[no head, from=3-11, to=5-13]
\end{tikzcd}\]

We keep substituting until there is nowhere else to sub. We do this number of nodes minus one times, which is equal to $l(F)+l(G)-3$, to get e.g.

\[\begin{tikzcd}[column sep=tiny,row sep=small]
{T_1} &&& {T_2} & {T_3} & {T_4} & {T_5} \\
\Delta & \Delta \\
& \Delta & \Delta \\
&& \Delta & \Delta & \mu \\
&&& \Delta & \mu & \mu \\
&&&&& \mu & \mu \\
&&&&&& \mu & \mu \\
&&&&&&& \mu & \mu \\
& {S_1} & {S_2} & {S_3} & {S_4} & {S_5} &&& {S_6}
\arrow[no head, from=1-5, to=3-2]
\arrow[no head, from=1-5, to=3-3]
\arrow[no head, from=3-2, to=4-3]
\arrow[no head, from=3-3, to=4-4]
\arrow[no head, from=4-3, to=1-6]
\arrow[no head, from=4-4, to=1-6]
\arrow[no head, from=4-5, to=5-4]
\arrow[no head, from=4-3, to=5-4]
\arrow[no head, from=4-5, to=5-6]
\arrow[squiggly, no head, from=4-4, to=5-5]
\arrow[no head, from=5-5, to=6-6]
\arrow[no head, from=5-6, to=6-7]
\arrow[no head, from=6-6, to=9-4]
\arrow[no head, from=5-5, to=9-3]
\arrow[no head, from=5-6, to=9-3]
\arrow[no head, from=6-7, to=9-4]
\arrow[no head, from=3-2, to=2-1]
\arrow[no head, from=3-3, to=2-2]
\arrow[no head, from=2-2, to=1-1]
\arrow[no head, from=2-2, to=1-4]
\arrow[no head, from=2-1, to=1-4]
\arrow[no head, from=4-5, to=1-7]
\arrow[no head, from=5-4, to=9-2]
\arrow[no head, from=2-1, to=1-1]
\arrow[no head, from=6-6, to=7-7]
\arrow[no head, from=6-7, to=7-8]
\arrow[no head, from=7-7, to=8-8]
\arrow[no head, from=7-8, to=8-9]
\arrow[no head, from=7-7, to=9-5]
\arrow[no head, from=7-8, to=9-5]
\arrow[no head, from=8-8, to=9-6]
\arrow[no head, from=8-9, to=9-6]
\arrow[no head, from=8-8, to=9-9]
\arrow[no head, from=8-9, to=9-9]
\end{tikzcd}\]

\[\begin{tikzcd}
{T_1} && {T_2} && {T_3} && {T_4} \\
\mu && \mu && \mu && \mu \\
\mu && \mu && \mu && \mu \\
\\
\\
\\
&&&&& \Delta && \Delta & \Delta \\
& \Delta & \Delta & \Delta & \Delta &&& \Delta \\
\Delta \\
{S_1} &&& {S_2} &&& {S_3}
\arrow[no head, from=2-1, to=1-1]
\arrow[no head, from=2-3, to=1-3]
\arrow[no head, from=2-5, to=1-5]
\arrow[no head, from=8-4, to=10-4]
\arrow[no head, from=2-1, to=8-8]
\arrow[no head, from=8-3, to=8-2]
\arrow[no head, from=8-8, to=10-7]
\arrow[no head, from=2-3, to=7-8]
\arrow[no head, from=3-7, to=8-3]
\arrow[no head, from=7-8, to=8-8]
\arrow[no head, from=2-7, to=7-9]
\arrow[no head, from=2-5, to=7-9]
\arrow[no head, from=2-7, to=3-7]
\arrow[no head, from=2-7, to=1-7]
\arrow[no head, from=3-1, to=9-1]
\arrow[no head, from=3-1, to=8-4]
\arrow[no head, from=8-2, to=9-1]
\arrow[no head, from=9-1, to=10-1]
\arrow[no head, from=3-1, to=2-1]
\arrow[no head, from=7-8, to=7-9]
\arrow[no head, from=3-3, to=8-5]
\arrow[no head, from=3-3, to=8-2]
\arrow[no head, from=3-3, to=2-3]
\arrow[no head, from=8-5, to=8-4]
\arrow[draw=none, from=3-5, to=7-6]
\arrow[no head, from=3-5, to=8-3]
\arrow[no head, from=3-7, to=7-6]
\arrow[no head, from=3-5, to=2-5]
\arrow[no head, from=7-6, to=8-5]
\end{tikzcd}\]

\[\begin{tikzcd}
& {T_1} &&& {T_2} & {T_3} & {T_4} & {T_5} \\
\Delta & \Delta \\
& \Delta & \Delta \\
&& \Delta & \mu & \mu & \mu \\
&&& \Delta & \Delta & \Delta & \mu \\
&&&&&& \mu & \mu \\
&&&&&&& \mu & \mu \\
&&&&&&&& \mu & \mu \\
&& {S_1} & {S_2} & {S_3} & {S_4} & {S_5} &&& {S_6}
\arrow[no head, from=1-6, to=3-2]
\arrow[no head, from=1-6, to=3-3]
\arrow[no head, from=3-2, to=4-3]
\arrow[no head, from=4-3, to=1-7]
\arrow[no head, from=4-5, to=1-7]
\arrow[no head, from=4-6, to=5-4]
\arrow[no head, from=4-3, to=5-4]
\arrow[no head, from=4-6, to=5-7]
\arrow[squiggly, no head, from=5-6, to=6-7]
\arrow[no head, from=5-7, to=6-8]
\arrow[no head, from=6-7, to=9-5]
\arrow[no head, from=5-7, to=9-4]
\arrow[no head, from=6-8, to=9-5]
\arrow[no head, from=3-2, to=2-1]
\arrow[no head, from=3-3, to=2-2]
\arrow[no head, from=2-2, to=1-2]
\arrow[no head, from=2-2, to=1-5]
\arrow[no head, from=2-1, to=1-5]
\arrow[no head, from=4-6, to=1-8]
\arrow[no head, from=5-4, to=9-3]
\arrow[no head, from=2-1, to=1-2]
\arrow[no head, from=6-7, to=7-8]
\arrow[no head, from=6-8, to=7-9]
\arrow[no head, from=7-8, to=8-9]
\arrow[no head, from=7-9, to=8-10]
\arrow[no head, from=7-8, to=9-6]
\arrow[no head, from=7-9, to=9-6]
\arrow[no head, from=8-9, to=9-7]
\arrow[no head, from=8-10, to=9-7]
\arrow[no head, from=8-9, to=9-10]
\arrow[no head, from=8-10, to=9-10]
\arrow[no head, from=4-4, to=5-6]
\arrow[no head, from=4-5, to=5-5]
\arrow[no head, from=4-4, to=5-5]
\arrow[no head, from=4-5, to=5-6]
\arrow[squiggly, no head, from=3-3, to=4-4]
\arrow[no head, from=5-5, to=9-4]
\end{tikzcd}\]

There is one place in the centre we can still perform a sub.

The number of times we have to do this will be the number of nodes minus one, which is equal to $l(F)+l(G)-3$. After doing this, we are left with a string diagram in which no multiplication comes before a comultiplication, which is what we need. However, the $\beta$ are in the wrong order. This is resolved by ... $\beta$-multiplication naturality says

\[\begin{tikzcd}[column sep=tiny]
& {T_1} &&&&& {T_2} &&&& {T_2} &&&&&& {T_1} \\
&&&&&&&&& {S_4} && {S_5} && {\,} && {S_1} & {S_2} & {S_3} \\
& {T_1} &&&&& {T_2} && {=} \\
{S_1} & {S_2} & {S_3} &&& {S_4} && {S_5} && {S_1} & {S_2} & {S_3} && {\qquad \qquad \qquad } && {S_4} && {S_5}
\arrow[no head, from=3-2, to=4-1]
\arrow[no head, from=3-2, to=4-2]
\arrow[no head, from=3-2, to=4-3]
\arrow[no head, from=4-6, to=3-7]
\arrow[no head, from=4-8, to=3-7]
\arrow[no head, from=3-2, to=1-7]
\arrow[no head, from=3-7, to=1-2]
\arrow[no head, from=2-12, to=1-11]
\arrow[no head, from=2-10, to=1-11]
\arrow[no head, from=2-16, to=1-17]
\arrow[no head, from=2-17, to=1-17]
\arrow[no head, from=2-18, to=1-17]
\arrow[no head, from=4-12, to=2-18]
\arrow[no head, from=4-11, to=2-17]
\arrow[no head, from=4-10, to=2-16]
\arrow[no head, from=4-18, to=2-12]
\arrow[no head, from=4-16, to=2-10]
\end{tikzcd}\]

\[\begin{tikzcd}[column sep=tiny,row sep=small]
&& I &&&&&& I \\
&&&&& {=} && {S_1 S_2} \\
{S_1} && {S_2} && {S_3} && {S_1} && {S_2} & {\ \ \ \ \ \ } & {S_3}
\arrow[no head, from=1-3, to=3-1]
\arrow[no head, from=1-3, to=3-3]
\arrow[no head, from=1-3, to=3-5]
\arrow[no head, from=3-7, to=2-8]
\arrow[no head, from=2-8, to=3-9]
\arrow[no head, from=2-8, to=1-9]
\arrow[no head, from=1-9, to=3-11]
\end{tikzcd}\]