\documentclass{article}
\input{packages}
\input{def}

\pdfmapline{=rossbb rossbb <rossbb.pfb}
\DeclareSymbolFont{rossbb}{T1}{rossbb}{m}{n}
\DeclareMathSymbol{\bbepsilon}{\mathord}{rossbb}{`e}
\DeclareMathSymbol{\bbsemicolon}{\mathpunct}{rossbb}{`;}

\lecture{Productors}

\begin{document}
\maketitle

\begin{definition}[Productor of an Effector $\langle E, \xmapsto{\bbsemicolon}, \noprf, \noprf \rangle$ for a 2-Category $\cat{C}$]
A tuple $\langle \ob{C}, \mo{m}, \mu, \prf{a}, \prf{i} \rangle$ whose components have the following types:
\begin{framed}
\begin{description}
\item[Object $\ob{C}$] is an object of $\cat{C}$
\item[Morphisms $\mo{m}$] maps each $\varepsilon : E$ to a morphism $\mo{m}_\varepsilon$ of $\cat{C}$ from $\ob{C}$ to $\ob{C}$
\item[Join $\mu$] maps each related pair $[\varepsilon_1, \dots, \varepsilon_n] \xmapsto{\bbsemicolon} \varepsilon$ to a 2-cell~$\mu_{[\varepsilon_1, \dots, \varepsilon_n]}^\varepsilon$ of $\cat{C}$ from $\mo{m}_{\varepsilon_1} \cocomp \dots \cocomp \mo{m}_{\varepsilon_n} \nto \mo{m}_\varepsilon$
\item[Associativity $\prf{a}$] proves
\begin{tikzpicture}[baseline=0]
\node(c1) at (0,1) {$\ob{C}$};
\node(c2) at (1,-1) {$\ob{C}$};
\node(c23) at (2,-1) {$\dots$};
\node(c3) at (3,-1) {$\ob{C}$};
\node(c4) at (4,1) {$\ob{C}$};
\draw[->] (c1) to[bend right=30] node[below,sloped]{$\mo{m}_{\varepsilon_1} \cocomp \dots \cocomp \mo{m}_{\varepsilon_m}$} node[near start](m11){} (c2);
\draw[->] (c2) to[bend right=30] node(m21){} (c23);
\draw[->] (c23) to[bend right=30] node(m31){} (c3);
\draw[->] (c3) to[bend right=30] node[below,sloped]{$\mo{m}_{\varepsilon_1}' \cocomp \dots \cocomp \mo{m}_{\varepsilon_n}'$} node[near start](m41){} (c4);
\draw[->] (c1) to[bend left=30] node[above,sloped]{$\mo{m}_\varepsilon$} node[near start](m12){} (c2);
\draw[->] (c2) to[bend left=30] node(m22){} (c23);
\draw[->] (c23) to[bend left=30] node(m32){} (c3);
\draw[->] (c3) to[bend left=30] node[above,sloped]{$\mo{m}_{\varepsilon'}$} node[near start](m42){} (c4);
\draw[->] (c1) -- node(f123){} node[above]{$\mo{m}_\epsilon$} (c4);
\draw[double,double equal sign distance,-implies] (m11) -- node[right,sloped,rotate=270,scale=.75]{$\mu_{[\varepsilon_1, \dots, \varepsilon_m]}^\varepsilon$} (m12);
\draw[double,double equal sign distance,-implies] (m21) -- (m22);
\draw[double,double equal sign distance,-implies] (m31) -- (m32);
\draw[double,double equal sign distance,-implies] (m41) -- node[right,sloped,rotate=90,scale=.75]{$\mu_{[\varepsilon_1, \dots, \varepsilon_n]}^\varepsilon$} (m42);
\draw[double,double equal sign distance,-implies] (c23) -- node[above,sloped]{$\mu_{[\varepsilon, \dots, \varepsilon']}^\epsilon$} (f123);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=0]
\node(c1) at (0,1) {$\ob{C}$};
\node(c2) at (1,-1) {$\ob{C}$};
\node(c23) at (2,-1) {$\dots$};
\node(c3) at (3,-1) {$\ob{C}$};
\node(c4) at (4,1) {$\ob{C}$};
\draw[->] (c1) to[bend right=30] node[below,sloped]{$\mo{m}_{\varepsilon_1} \cocomp \dots \cocomp \mo{m}_{\varepsilon_m}$} node[near start](m11){} (c2);
\draw[->] (c2) to[bend right=30] node(m21){} (c23);
\draw[->] (c23) to[bend right=30] node(m31){} (c3);
\draw[->] (c3) to[bend right=30] node[below,sloped]{$\mo{m}_{\varepsilon_1}' \cocomp \dots \cocomp \mo{m}_{\varepsilon_n}'$} node[near start](m41){} (c4);
\draw[->] (c1) -- node(f123){} node[above]{$\mo{m}_\epsilon$} (c4);
\draw[double,double equal sign distance,-implies] (c23) -- node[above,sloped,scale=.65]{$\mu_{[\varepsilon_1,\dots,\varepsilon_n, \dots, \varepsilon_1',\dots,\varepsilon_n']}^\epsilon$} (f123);
\end{tikzpicture}
.\\
In other words,
\begin{tikzpicture}[baseline=1.5cm]
\node(f11) at (0,0) {$\mo{m}_{\varepsilon_1}$};
\node(f12) at (.5,0) {$\dots$};
\node(f13) at (1,0) {$\mo{m}_{\varepsilon_m}$};
\node(f2) at (2,0) {$\dots$};
\node(f31) at (3,0) {$\mo{m}_{\varepsilon_1'}$};
\node(f32) at (3.5,0) {$\dots$};
\node(f33) at (4,0) {$\mo{m}_{\varepsilon_n'}$};
\node(n1)[circle,draw] at (.5,1) {$\mu$};
\node(n2) at (2,1) {$\dots$};
\node(n3)[circle,draw] at (3.5,1) {$\mu$};
\node(n)[circle,draw] at (2,2) {$\mu$};
\node(f123) at (2,3) {$\mo{m}$};
\draw[->] (f11) -- (n1);
\draw[->] ($(f12.north)+(-.05,0)$) -- ($(n1.south)+(-.05,0)$);
\draw[->] (f12) -- (n1);
\draw[->] ($(f12.north)+(.05,0)$) -- ($(n1.south)+(.05,0)$);
\draw[->] (f13) -- (n1);
\draw[->] ($(f2.north)+(-.1,0)$) -- ($(n2.south)+(-.1,0)$);
\draw[->] ($(f2.north)+(-.05,0)$) -- ($(n2.south)+(-.05,0)$);
\draw[->] (f2) -- (n2);
\draw[->] ($(f2.north)+(.05,0)$) -- ($(n2.south)+(.05,0)$);
\draw[->] ($(f2.north)+(.1,0)$) -- ($(n2.south)+(.1,0)$);
\draw[->] (f31) -- (n3);
\draw[->] ($(f32.north)+(-.05,0)$) -- ($(n3.south)+(-.05,0)$);
\draw[->] (f32) -- (n3);
\draw[->] ($(f32.north)+(.05,0)$) -- ($(n3.south)+(.05,0)$);
\draw[->] (f33) -- (n3);
\draw[->] (n1) -- node[above left]{$\mo{m}_\varepsilon$} (n);
\draw[->] ($(n2.north)+(-.05,0)$) -- ($(n.south)+(-.05,0)$);
\draw[->] (n2) -- (n);
\draw[->] ($(n2.north)+(.05,0)$) -- ($(n.south)+(.05,0)$);
\draw[->] (n3) -- node[above right]{$\mo{m}_{\varepsilon'}$} (n);
\draw[->] (n) -- (f123);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=1.5cm]
\node(f11) at (0,0) {$\mo{m}_{\varepsilon_1}$};
\node(f12) at (.5,0) {$\dots$};
\node(f13) at (1,0) {$\mo{m}_{\varepsilon_m}$};
\node(f2) at (2,0) {$\dots$};
\node(f31) at (3,0) {$\mo{m}_{\varepsilon_1'}$};
\node(f32) at (3.5,0) {$\dots$};
\node(f33) at (4,0) {$\mo{m}_{\varepsilon_n'}$};
\node(n)[circle,draw] at (2,2) {$\mu$};
\node(f123) at (2,3) {$\mo{m}$};
\draw[->] (f11) -- (n.215);
\draw[->] ($(f12.north)+(-.15,0)$) -- (n.220);
\draw[->] ($(f12.north)+(.05,0)$) -- (n.225);
\draw[->] ($(f12.north)+(.25,0)$) -- (n.230);
\draw[->] (f13) -- (n.235);
\draw[->] ($(f2.north)+(-.1,0)$) -- ($(n.south)+(-.1,0)$);
\draw[->] ($(f2.north)+(-.05,0)$) -- ($(n.south)+(-.05,0)$);
\draw[->] (f2) -- (n);
\draw[->] ($(f2.north)+(.05,0)$) -- ($(n.south)+(.05,0)$);
\draw[->] ($(f2.north)+(.1,0)$) -- ($(n.south)+(.1,0)$);
\draw[->] (f31) -- (n.305);
\draw[->] ($(f32.north)+(-.25,0)$) -- (n.310);
\draw[->] ($(f32.north)+(-.05,0)$) -- (n.315);
\draw[->] ($(f32.north)+(.15,0)$) -- (n.320);
\draw[->] (f33) -- (n.325);
\draw[->] (n) -- (f123);
\end{tikzpicture}
.
\item[Identity $\prf{i}$] is a proof that $\forall \varepsilon : E.\; \mu_{[\varepsilon]}^\varepsilon = \id_{m_\varepsilon} : m_\varepsilon \nto m_\varepsilon$
\end{description}
\end{framed}
\end{definition}

\begin{exercise}
Prove that a monad is a productor for the effector with one element and with $\xmapsto{\bbsemicolon}$ always true.
\end{exercise}

\begin{exercise}
Every effector corresponds to a thin multicategory, which in turn corresponds to an opetory with 1 object and at most one 2-cell from any domain to any codomain.
Prove that a productor for an effector is simply a functor from the corresponding opetory.
\end{exercise}

\begin{definition}[Productoid of an Effectoid $\langle E, \bbepsilon \mapsto \bullet, \leq, \bullet \mathop{\bbsemicolon} \bullet \mapsto \bullet, \noprf \rangle$ for a 2-Category $\cat{C}$]
A tuple $\langle \ob{C}, \mo{m}, \mu_\bbepsilon, \mu_\leq, \mu_\bbsemicolon, \prf{c} \rangle$ whose components have the following types:
\begin{framed}
\begin{description}
\item[Object $\ob{C}$] is an object of $\cat{C}$
\item[Morphisms $\mo{m}$] maps each $\varepsilon : E$ to a morphism $\mo{m}_\varepsilon$ of $\cat{C}$ from $\ob{C}$ to $\ob{C}$
\item[Unit $\mu_\bbepsilon$] maps each $\varepsilon : E$ satisfying $\bbepsilon \mapsto \varepsilon$ to a 2-cell~$\mu_\bbepsilon^\varepsilon$ of $\cat{C}$ from $\id_\ob{C} \nto \mo{m}_\varepsilon$
\item[Coercion $\mu_\leq$] maps each $\varepsilon, \varepsilon' : E$ satisfying $\varepsilon \leq \varepsilon'$ to a 2-cell~$\mu_{\varepsilon \leq}^{\varepsilon'}$ of $\cat{C}$ from $\mo{m}_\varepsilon \nto \mo{m}_{\varepsilon'}$
\item[Join $\mu_\bbsemicolon$] maps each $\varepsilon_1, \varepsilon_2, \varepsilon : E$ satisfying $\varepsilon_1 \bbsemicolon \varepsilon_2 \mapsto \varepsilon$ to a 2-cell~$\mu_{\varepsilon_1 \bbsemicolon \varepsilon_2}^\varepsilon$ of $\cat{C}$ from $\mo{m}_{\varepsilon_1} \cocomp \mo{m}_{\varepsilon_2} \nto \mo{m}_\varepsilon$
\item[Coherence $\prf{c}$] is a proof that $\mu_{\varepsilon \leq}^\varepsilon$ always equals $\id_{\mo{m}_\varepsilon}$ and the following equalities all hold whenever well defined:\\\
First,
\begin{tikzpicture}[baseline=(c1.base)]
\node(c1) at (0,0) {$\ob{C}$};
\node(c2) at (4,0) {$\ob{C}$};
\draw[->] (c1) .. controls (1.5,-1.5) and (2.5,-1.5) .. node(f1){} node[below]{$\mo{m}_\varepsilon$} (c2);
\draw[->] (c2) .. controls (1.75,-1.75) and (1.75,1.75) .. node(f2){} node[left]{$\mo{m}_{\varepsilon_r}$} (c2);
\draw[->] (c1) .. controls (1.5,1.5) and (2.5,1.5) .. node(f12){} node[above]{$\mo{m}_{\varepsilon'}$} (c2);
\draw[double,double equal sign distance,-implies] (c2) -- node[above]{$\mu_\bbepsilon^{\varepsilon_r}$} (f2);
\draw[double,double equal sign distance,-implies] ($(f1.west) !.5! (c2.south)$) .. controls +(-2,-.5) and ($(f12.south west) + (-1,-1)$) .. node[above left]{$\mu_{\varepsilon \bbsemicolon \varepsilon_r}^{\varepsilon'}$} (f12.south west);
\end{tikzpicture}
and
\begin{tikzpicture}[baseline=(c1.base)]
\node(c1) at (0,0) {$\ob{C}$};
\node(c2) at (4,0) {$\ob{C}$};
\draw[->] (c1) .. controls (2.25,1.75) and (2.25,-1.75) .. node(f1){} node[right]{$\mo{m}_{\varepsilon_\ell}$} (c1);
\draw[->] (c1) .. controls (1.5,-1.5) and (2.5,-1.5) .. node(f2){} node[below]{$\mo{m}_\varepsilon$} (c2);
\draw[->] (c1) .. controls (1.5,1.5) and (2.5,1.5) .. node(f12){} node[above]{$\mo{m}_{\varepsilon'}$} (c2);
\draw[double,double equal sign distance,-implies] (c1) -- node[above]{$\mu_\bbepsilon^{\varepsilon_\ell}$} (f1);
\draw[double,double equal sign distance,-implies] ($(f2.east) !.5! (c1.south)$) .. controls +(2,-.5) and ($(f12.south east) + (1,-1)$) .. node[above right]{$\mu_{\varepsilon_\ell \bbsemicolon \varepsilon}^{\varepsilon'}$} (f12.south east);
\end{tikzpicture}
equal
\begin{tikzpicture}[baseline=(c1.base)]
\node(c1) at (0,0) {$\ob{C}$};
\node(c2) at (2,0) {$\ob{C}$};
\draw[->] (c1) to[bend right=30] node[below]{$\mo{m}_\varepsilon$} node(f1)[near start]{} (c2);
\draw[->] (c1) to[bend left=30] node[above]{$\mo{m}_{\varepsilon'}$} node(f2)[near start]{} (c2);
\draw[double,double equal sign distance,-implies] (f1) -- node[right]{$\mu_{\varepsilon \leq}^{\varepsilon'}$} (f2);
\end{tikzpicture}
.\\
In other words,
\begin{tikzpicture}[baseline=1.5cm]
\node(f1) at (0,0) {$\mo{m}_\varepsilon$};
\node(n1)[circle,draw] at (1,1) {$\mu$};
\node(n2)[circle,draw] at (0,2) {$\mu$};
\node(f12) at (0,3) {$\mo{m}_{\varepsilon'}$};
\draw[->] (f1) -- ++(0,.5) .. controls +(0,.5) and ($(n2.south west) + (-.5,-.5)$) .. (n2.south west);
\draw[->] (n1) -- node[above right]{$\mo{m}_{\varepsilon_r}$} (n2);
\draw[->] (n2) -- (f12);
\node at (-1,1) {$\ob{C}$};
\node at (1.5,2.25) {$\ob{C}$};
\end{tikzpicture}
and
\begin{tikzpicture}[baseline=1.5cm]
\node(f1) at (0,0) {$\mo{m}_\varepsilon$};
\node(n1)[circle,draw] at (-1,1) {$\mu$};
\node(n2)[circle,draw] at (0,2) {$\mu$};
\node(f12) at (0,3) {$\mo{m}_{\varepsilon'}$};
\draw[->] (f1) -- ++(0,.5) .. controls +(0,.5) and ($(n2.south east) + (.5,-.5)$) .. (n2.south east);
\draw[->] (n1) -- node[above left]{$\mo{m}_{\varepsilon_\ell}$} (n2);
\draw[->] (n2) -- (f12);
\node at (1,1) {$\ob{C}$};
\node at (-1.5,2.25) {$\ob{C}$};
\end{tikzpicture}
equal
\begin{tikzpicture}[baseline=1.5cm]
\node(f1) at (0,0) {$\mo{m}_\varepsilon$};
\node(n)[circle,draw] at (0,1.5) {$\mu$};
\node(f2) at (0,3) {$\mo{m}_{\varepsilon'}$};
\draw[->] (f1) -- (n);
\draw[->] (n) -- (f2);
\node at (-1,1.5) {$\ob{C}$};
\node at (1,1.5) {$\ob{C}$};
\end{tikzpicture}
.\\
Second,
\begin{tikzpicture}[baseline=0]
\node(c1) at (0,1) {$\ob{C}$};
\node(c2) at (1,-1) {$\ob{C}$};
\node(c3) at (3,-1) {$\ob{C}$};
\node(c4) at (4,1) {$\ob{C}$};
\draw[->] (c1) -- node[below left]{$\mo{m}_{\varepsilon_1}$} (c2);
\draw[->] (c2) -- node[below]{$\mo{m}_{\varepsilon_2}$} (c3);
\draw[->] (c3) -- node[below right]{$\mo{m}_{\varepsilon_3}$} (c4);
\draw[->] (c1) -- node(f12){} node[above]{$\mo{m}_{\varepsilon_{12}}$} (c3);
\draw[->] (c1) -- node(f123){} node[above]{$\mo{m}_{\varepsilon_{123}}$} (c4);
\draw[double,double equal sign distance,-implies] (c2) -- node[below right]{$\mu_{\varepsilon_1 \bbsemicolon \varepsilon_2}^{\varepsilon_{12}}$} (f12);
\draw[double,double equal sign distance,-implies] (c3) -- node[above right]{$\mu_{\varepsilon_{12} \bbsemicolon \varepsilon_3}^{\varepsilon_{123}}$} (f123);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=0]
\node(c1) at (0,1) {$\ob{C}$};
\node(c2) at (1,-1) {$\ob{C}$};
\node(c3) at (3,-1) {$\ob{C}$};
\node(c4) at (4,1) {$\ob{C}$};
\draw[->] (c1) -- node[below left]{$\mo{m}_{\varepsilon_1}$} (c2);
\draw[->] (c2) -- node[below]{$\mo{m}_{\varepsilon_2}$} (c3);
\draw[->] (c3) -- node[below right]{$\mo{m}_{\varepsilon_3}$} (c4);
\draw[->] (c2) -- node(f23){} node[above]{$\mo{m}_{\varepsilon_{23}}$} (c4);
\draw[->] (c1) -- node(f123){} node[above]{$\mo{m}_{\varepsilon_{123}}$} (c4);
\draw[double,double equal sign distance,-implies] (c3) -- node[below left]{$\mu_{\varepsilon_2 \bbsemicolon \varepsilon_3}^{\varepsilon_{23}}$} (f23);
\draw[double,double equal sign distance,-implies] (c2) -- node[above left]{$\mu_{\varepsilon_1 \bbsemicolon \varepsilon_{23}}^{\varepsilon_{123}}$} (f123);
\end{tikzpicture}
.\\
In other words,
\begin{tikzpicture}[baseline=1.5cm]
\node(f1) at (0,0) {$\mo{m}_{\varepsilon_1}$};
\node(f2) at (2,0) {$\mo{m}_{\varepsilon_2}$};
\node(f3) at (4,0) {$\mo{m}_{\varepsilon_3}$};
\node(n1)[circle,draw] at (1,1) {$\mu$};
\node(n2)[circle,draw] at (2,2) {$\mu$};
\node(f123) at (2,3) {$\mo{m}_{\varepsilon_{123}}$};
\draw[->] (f1) -- (n1);
\draw[->] (f2) -- (n1);
\draw[->] (f3) -- (n2);
\draw[->] (n1) -- node(f12)[above left]{$\mo{m}_{\varepsilon_{12}}$} (n2);
\draw[->] (n2) -- (f123);
\node at (.5,2.5) {$\ob{C}$};
\node at (1,.25) {$\ob{C}$};
\node at (2.25,.75) {$\ob{C}$};
\node at (3.5,2) {$\ob{C}$};
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=1.5cm]
\node(f1) at (0,0) {$\mo{m}_{\varepsilon_1}$};
\node(f2) at (2,0) {$\mo{m}_{\varepsilon_2}$};
\node(f3) at (4,0) {$\mo{m}_{\varepsilon_3}$};
\node(n1)[circle,draw] at (3,1) {$\mu$};
\node(n2)[circle,draw] at (2,2) {$\mu$};
\node(f123) at (2,3) {$\mo{m}_{\varepsilon_{123}}$};
\draw[->] (f1) -- (n2);
\draw[->] (f2) -- (n1);
\draw[->] (f3) -- (n1);
\draw[->] (n1) -- node(f12)[above right]{$\mo{m}_{\varepsilon_{23}}$} (n2);
\draw[->] (n2) -- (f123);
\node at (3.5,2.5) {$\ob{C}$};
\node at (3,.25) {$\ob{C}$};
\node at (1.75,.75) {$\ob{C}$};
\node at (.5,2) {$\ob{C}$};
\end{tikzpicture}
.\\
Third, $\id_\ob{C} \xRightarrow{\mu_\bbepsilon^\varepsilon\,} \mo{m}_\varepsilon \xRightarrow{\mu_{\varepsilon \leq}^{\varepsilon'}\,} \mo{m}_{\varepsilon'}$ equals $\id_\ob{C} \xRightarrow{\mu_\bbepsilon^{\varepsilon'}} \mo{m}_{\varepsilon'}$.\\
Fourth, $\mo{m}_{\varepsilon_1} \cocomp \mo{m}_{\varepsilon_2} \xRightarrow{\mu_{\varepsilon_1 \bbsemicolon \varepsilon_2}^\varepsilon\,} \mo{m}_\varepsilon \xRightarrow{\mu_{\varepsilon \leq}^{\varepsilon'}} \mo{m}_{\varepsilon'}$ equals $\mo{m}_{\varepsilon_1} \cocomp \mo{m}_{\varepsilon_2} \xRightarrow{\mu_{\varepsilon_1 \bbsemicolon \varepsilon_2}^{\varepsilon'}\,} \mo{m}_{\varepsilon'}$.
\vspace{-.2in}
\end{description}
\end{framed}
\end{definition}

\begin{theorem}
If an effector is semi-strict, then there is a bijection between the set of productors for that effector and the set of productoids for the effectoid corresponding to that semi-strict effector.
The bijection preserves the object~$\ob{C}$ and the mapping~$\mo{m}$.
The 2-cells $\mu_\bbepsilon^\varepsilon$ correspond to the 2-cells $\mu_{\nil}^\varepsilon$; the 2-cells $\mu_{\varepsilon \leq}^{\varepsilon'}$ correspond to the 2-cells $\mu_{[\varepsilon]}^{\varepsilon'}$; and the 2-cells $\mu_{\varepsilon_1 \bbsemicolon \varepsilon_2}^\varepsilon$ correspond to the 2-cells $\mu_{[\varepsilon_1, \varepsilon_2]}^\varepsilon$.
\end{theorem}

\begin{definition}[Postmodule of a Productor]
A tuple $\langle \ob{R}, \mo{r}, \rho, \prf{d} \rangle$ whose components have the following types:
\begin{framed}\vspace{-.15in}
\begin{description}
\item[Object $\ob{R}$:] $\cat{C}$
\item[Morphism $\mo{r}$:] $\ob{C} \mto \ob{R}$
\item[Action $\rho$:] A mapping from each $\varepsilon : E$ to a 2-cell $\rho_\varepsilon : \mo{m}_\varepsilon \cocomp \mo{r} \nto \mo{r}$
\item[Distributivity $\prf{d}$:] A proof that
\begin{tikzpicture}[baseline=(r.base)]
\node(c1) at (2,2) {$\ob{C}$};
\node(c2) at (0,1) {$\ob{C}$};
\node(c3) at (1,-.5) {$\ob{C}$};
\node(c4) at (3,-.5) {$\ob{C}$};
\node(r) at (4,1) {$\ob{R}$};
\draw[->] (c1) -- node[above left]{$\mo{m}_{\varepsilon_1}$} (c2);
\draw[->] (c2) -- node[below left]{$\dots$} (c3);
\draw[->] (c3) -- node[below]{$\mo{m}_{\varepsilon_n}$} (c4);
\draw[->] (c4) -- node[below right]{$\mo{r}$} (r);
\draw[->] (c3) -- node(f3){} node[above]{$\mo{r}$} (r);
\draw[->] (c2) -- node(f2)[below]{$\mo{r}$} (r);
\draw[->] (c1) -- node(f1){} node[above right]{$\mo{r}$} (r);
\draw[double,double equal sign distance,-implies] (c4) -- node[below left]{$\rho_{\varepsilon_n}$} (f3);
\draw[double,double equal sign distance,-implies] (c3) -- node[above left]{$\dots$} (f2);
\draw[double,double equal sign distance,-implies,shorten <=3mm] (c2) -- node[above]{$\rho_{\varepsilon_1}$} (f1);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=(r.base)]
\node(c1) at (2,2) {$\ob{C}$};
\node(c2) at (0,1) {$\ob{C}$};
\node(c3) at (1,-.5) {$\ob{C}$};
\node(c4) at (3,-.5) {$\ob{C}$};
\node(r) at (4,1) {$\ob{R}$};
\draw[->] (c1) -- node[above left]{$\mo{m}_{\varepsilon_1}$} (c2);
\draw[->] (c2) -- node(m2){} node[below left]{$\dots$} (c3);
\draw[->] (c3) -- node[below]{$\mo{m}_{\varepsilon_n}$} (c4);
\draw[->] (c4) -- node[below right]{$\mo{r}$} (r);
\draw[->] (c1) -- node(f1){} node[above right]{$\mo{r}$} (r);
\draw[->] (c1) -- node(m)[left]{$\mo{m}_\varepsilon$} (c4);
\draw[double,double equal sign distance,-implies,shorten <=1mm] (m2) -- node[above,sloped]{$\mu_{[\varepsilon_1, \dots, \varepsilon_n]}^\varepsilon$} (m);
\draw[double,double equal sign distance,-implies,shorten <=3mm] (c4) -- node[right]{$\rho_{\varepsilon}$} (f1);
\end{tikzpicture}
.
\vspace{-.2in}
\end{description}
\end{framed}
\end{definition}

\end{document}