<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{A Denotational Model for Intensional Gradual Typing}

In this section we model intensional gradual typing in a suitable double category.

We construct a (thin) double category $\mathsf{IGTT}$ as follows.

% First, we fix a set $D$ with a partial order $\le_D$ and bisimilarity relation $\bisim_D$.
% This is intended to model the dynamic type. Now we define the double category:

\begin{itemize}
  \item \textbf{Objects}: An object consists of the following data:
    \begin{itemize}
        \item A double poset $X$, i.e., a set $X$ equipped with a partial order $\le_X$
        and a reflexive, symmetric ``bisimilarity'' relation $\bisim_X$.
        \item Two commutative monoids of perturbations $P_V$ and $P_C$ with homomorphisms
        \begin{align*}
        \ptb_V &amp;: P_V \to \{ f : X \to_m X \mid f \bisim \id \} \\
        \ptb_C &amp;: P_C \to \{ f : X \to_m \li X \mid f \bisim \eta \}
        \end{align*}
        (where composition in the latter monoid of functions is given by Kleisli composition).

    \end{itemize}

  \item \textbf{Vertical arrows}: An vertical arrow from $(X, P_V^X, P_C^X)$ to $(Y, P_V^Y, P_C^Y)$ is
  a function $f : X \to Y$ that is \emph{monotone} (preserves ordering) and preserves the bisimilarity relation.
  % that preserves ordering and bisimilarity.
  
  \item \textbf{Horizontal arrows}: A horizontal arrow from $(X, P_V^X, P_C^X)$ to $(Y, P_V^Y, P_C^Y)$
  consists of:
  \begin{itemize}
    \item A relation $R : X \nrightarrow Y$ that is antitone with respect to $\le_X$ and
    monotone with respect to $\le_Y$.
    \item An embedding $e_{XY} : X \to_m Y$ preserving ordering and bisimilarity.
    \item A projection $p_{XY} : Y \to_m \li X$ preserving ordering and bisimilarity.
  \end{itemize}
  
  such that (1) $R$ is \emph{quasi-representable} by $e_{XY}$ and $p_{XY}$, and 
  (2) $R$ satisfies the \emph{push-pull} property.
  
  \vspace{3ex}
  The former means that there are distinguished elements $\delta^{l,e} \in P_V^X$, $\delta^{r,e} \in P_V^Y$, 
  $\delta^{l,p} \in P_C^X$ and $\delta^{r,p} \in P_C^Y$ such that the following squares commute:

  \begin{center}
    \begin{tabular}{ c | c } 
        \hline
        \hspace{3em} 
        % UpL
        % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWSJdLFsxLDAsIlkiXSxbMSwxLCJZIl0sWzAsMiwiUiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzEsMywiXFxsZV9ZIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwxLCJlX3tYWX0iLDJdLFsyLDMsIlxccHRiX1ZeWShcXGRlbHRhXntyLGV9KSJdLFs2LDcsIlxcbHRkeW4iLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
        \begin{tikzcd}[ampersand replacement=\&amp;]
            X \&amp; Y \\
            Y \&amp; Y
            \arrow["R", "\shortmid"{marking}, no head, from=1-1, to=1-2]
            \arrow["{\le_Y}"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
            \arrow[""{name=0, anchor=center, inner sep=0}, "{e_{XY}}"', from=1-1, to=2-1]
            \arrow[""{name=1, anchor=center, inner sep=0}, "{\ptb_V^Y(\delta^{r,e})}", from=1-2, to=2-2]
            \arrow["\ltdyn"{description}, draw=none, from=0, to=1]
        \end{tikzcd} &amp; 
        % UpR
        % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWCJdLFsxLDAsIlgiXSxbMSwxLCJZIl0sWzAsMSwiXFxwdGJfVl5YKFxcZGVsdGFee2wsZX0pIiwyXSxbMiwzLCJlX3tYWX0iXSxbMSwzLCJSIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwyLCJcXGxlX1giLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs0LDUsIlxcbHRkeW4iLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
        \begin{tikzcd}[ampersand replacement=\&amp;]
            X \&amp; X \\
            X \&amp; Y
            \arrow[""{name=0, anchor=center, inner sep=0}, "{\ptb_V^X(\delta^{l,e})}"', from=1-1, to=2-1]
            \arrow[""{name=1, anchor=center, inner sep=0}, "{e_{XY}}", from=1-2, to=2-2]
            \arrow["R"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
            \arrow["{\le_X}", "\shortmid"{marking}, no head, from=1-1, to=1-2]
            \arrow["\ltdyn"{description}, draw=none, from=0, to=1]
        \end{tikzcd} \\
        \hline
        % DnR
        % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMCwiWSJdLFswLDEsIkxYIl0sWzEsMSwiTFgiXSxbMCwyLCJcXHB0Yl9DXlgoXFxkZWx0YV57bCxwfSkiLDJdLFsxLDMsInBfe1hZfSJdLFswLDEsIlIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFsyLDMsIlxcbGVfe0xYfSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzQsNSwiXFxsdGR5biIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
        \begin{tikzcd}[ampersand replacement=\&amp;]
            X \&amp; Y \\
            LX \&amp; LX
            \arrow[""{name=0, anchor=center, inner sep=0}, "{\ptb_C^X(\delta^{l,p})}"', from=1-1, to=2-1]
            \arrow[""{name=1, anchor=center, inner sep=0}, "{p_{XY}}", from=1-2, to=2-2]
            \arrow["R", "\shortmid"{marking}, no head, from=1-1, to=1-2]
            \arrow["{\le_{LX}}"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
            \arrow["\ltdyn"{description}, draw=none, from=0, to=1]
        \end{tikzcd} &amp; 
            \hspace{3em} 
            % DnL
            % https://q.uiver.app/#q=WzAsNCxbMCwwLCJZIl0sWzEsMCwiWSJdLFswLDEsIkxYIl0sWzEsMSwiTFkiXSxbMCwxLCJcXGxlX1kiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFswLDIsInBfe1hZfSIsMl0sWzEsMywiXFxwdGJfQ15ZKFxcZGVsdGFee3IscH0pIl0sWzIsMywiTFIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs1LDYsIlxcbHRkeW4iLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
            \begin{tikzcd}[ampersand replacement=\&amp;]
                Y \&amp; Y \\
                LX \&amp; LY
                \arrow["{\le_Y}", "\shortmid"{marking}, no head, from=1-1, to=1-2]
                \arrow[""{name=0, anchor=center, inner sep=0}, "{p_{XY}}"', from=1-1, to=2-1]
                \arrow[""{name=1, anchor=center, inner sep=0}, "{\ptb_C^Y(\delta^{r,p})}", from=1-2, to=2-2]
                \arrow["LR"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
                \arrow["\ltdyn"{description}, draw=none, from=0, to=1]
            \end{tikzcd} \\ 
        \hline
    \end{tabular}
    \end{center}
    (Here, $LR$ is the lock step lifting of the relation $R$.)

    \vspace{3ex}

    The push-pull property is defined as follows:
    \begin{itemize}
        \item Given any perturbation $\delta_X \in P_V^X$, we can \emph{push} it forward along $R$ to a
        perturbation $\push(\delta_X) \in P_V^Y$, such that $\ptb_V^X(\delta_X) \le \ptb_V^Y(\push(\delta_X))$.

        \item Conversely, given any perturbation $\delta_Y \in P_V^Y$, we can \emph{pull} it back along $R$
        to a perturbation $\pull(\delta_Y) \in P_V^X$, such that $\ptb_V^X(\pull(\delta_Y)) \le \ptb_V^Y(\delta_Y)$.

        \item Likewise, we can push any perturbation $\delta_X \in P_C^X$ along $LR$
        to get a perturbation $\push(\delta_X) \in P_C^Y$ such that
        $\ptb_C^X(\delta_X) \le \ptb_C^Y(\push(\delta_X))$.

        \item And similarly, we can pull a perturbation in $P_C^Y$ along $LR$ to a perturbation in $P_C^X$
        satisfying the analogous property.
    \end{itemize}

    \textbf{TODO: push and pull might need to be monoid homomorphisms}

  \item \textbf{Two-cells}: Let $f : W \to X$ and $g : Y \to Z$ and let $R : W \nrightarrow Y$ and 
  $S : X \nrightarrow Z$. We define $f \le g$ to mean for all $(w, y) \in R$, we have
  $(f(w), g(y)) \in S$. This is depicted in the square below:

  % https://q.uiver.app/#q=WzAsNCxbMCwwLCJXIl0sWzAsMSwiWCJdLFsxLDAsIlkiXSxbMSwxLCJaIl0sWzAsMiwiUiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzEsMywiUyIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzAsMSwiZiIsMl0sWzIsMywiZyJdLFs0LDUsIlxcc3FzdWJzZXRlcSIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
\[\begin{tikzcd}[ampersand replacement=\&amp;]
	W \&amp; Y \\
	X \&amp; Z
	\arrow[""{name=0, anchor=center, inner sep=0}, "R", "\shortmid"{marking}, no head, from=1-1, to=1-2]
	\arrow[""{name=1, anchor=center, inner sep=0}, "S"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
	\arrow["f"', from=1-1, to=2-1]
	\arrow["g", from=1-2, to=2-2]
	\arrow["\sqsubseteq"{description}, draw=none, from=0, to=1]
\end{tikzcd}\]
  
\end{itemize}

The category satisfies the following additional properties:
\begin{itemize}
    \item \emph{Existence of Dyn}: There is an object $D$ with the property that for any
    object $X$, there is a horizontal arrow $X \nrightarrow D$.
    The underlying double poset is defined by guarded recursion as the solution to
    \[ D \cong \mathbb{N}\, + \later \hspace{-0.5ex} (D \to_m \li D). \]

    \textbf{TODO: define the perturbations for Dyn and show there is a horizontal arrow $X \nrightarrow D$ for all $X$.}
    
    \item \emph{Thinness}: There is at most one two-cell for any given square.

    % \item \emph{Push-Pull}: Let $X$ and $Y$ be objects, and let $R : X \nrightarrow Y$.
    % \begin{itemize}
    %     \item Given any perturbation $\delta_X \in P_V^X$, we can \emph{push} it forward along $R$ to a
    %     perturbation $\push(\delta_X) \in P_V^Y$, such that $\ptb_V^X(\delta_X) \le \ptb_V^Y(\push(\delta_X))$.

    %     \item Conversely, given any perturbation $\delta_Y \in P_V^Y$, we can \emph{pull} it back along $R$
    %     to a perturbation $\pull(\delta_Y) \in P_V^X$, such that $\ptb_V^X(\pull(\delta_Y)) \le \ptb_V^Y(\delta_Y)$.

    %     \item Likewise, we can push any perturbation $\delta_X \in P_C^X$ along $L\, R$
    %     (the lock-step lifting of the relation $R$) to get a perturbation $\push(\delta_X) \in P_C^Y$ such that
    %     $\ptb_C^X(\delta_X) \le \ptb_C^Y(\push(\delta_X))$.

    %     \item And similarly we can pull a perturbation in $P_C^Y$ to a perturbation in $P_C^X$
    %     satisfying the analogous property.
    % \end{itemize}
\end{itemize}

% Composability of embedding and projections

We need to verify that this forms a thin double category. 
\begin{itemize}
    \item \emph{Horizontal identity morphism}: 
    Let $X$ be an object. We take $R$ to be $\le_X$ (the ordering relation on $X$),
    which is trivially antitone and monotone with respect to itself.
    We let $e_{XX} = \id$ and $p_{XX} = \eta$. These clearly preserve the
    ordering and bisimilarity.

    \vspace{3ex}
    
    We first need to show that $R$ is quasi-representable.
    We prove the UpR rule; the others are similar.
    We need to specify a distinguished element $\delta^{l,e} \in P_V^X$ such that
    the following square commutes:
    
    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWCJdLFsxLDAsIlgiXSxbMSwxLCJYIl0sWzAsMSwiXFxwdGJfVl5YKFxcZGVsdGFee2wsZX0pIiwyXSxbMiwzLCJlX3tYWH0gPSBcXGlkIl0sWzAsMiwiXFxsZV9YIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMSwzLCJcXGxlX1giLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
    \[\begin{tikzcd}[ampersand replacement=\&amp;]
        X \&amp; X \\
        X \&amp; X
        \arrow["{\ptb_V^X(\delta^{l,e})}"', from=1-1, to=2-1]
        \arrow["{e_{XX} = \id}", from=1-2, to=2-2]
        \arrow["{\le_X}", "\shortmid"{marking}, no head, from=1-1, to=1-2]
        \arrow["{\le_X}"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
    \end{tikzcd}\]

    Taking $\delta^{l,e} = \id$ (the identity of the monoid), we observe that since
    $\ptb_V^X$ is a homomorphism of monoids, we have $\ptb_V^X(\id) = \id$.
    Now it is clear that the above square commutes.

    \vspace{3ex}

    We also need to show that $R$ satisfies the four push-pull properties.
    We show one; the others are similar. Let $\delta_X \in P_V^X$.
    We need to define $\push(\delta_X) \in P_V^X$ such that the following square commutes:

    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWCJdLFsxLDAsIlgiXSxbMSwxLCJYIl0sWzAsMiwiXFxsZV9YIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMSwzLCJcXGxlX1giLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFswLDEsIlxccHRiX1ZeWChcXGRlbHRhX1gpIiwyXSxbMiwzLCJcXHB0Yl9WXlgoXFxwdXNoKFxcZGVsdGFfWCkpIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d
    \[\begin{tikzcd}[ampersand replacement=\&amp;]
	X \&amp; X \\
	X \&amp; X
	\arrow["{\le_X}", "\shortmid"{marking}, no head, from=1-1, to=1-2]
	\arrow["{\le_X}"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
	\arrow["{\ptb_V^X(\delta_X)}"', from=1-1, to=2-1]
	\arrow["{\ptb_V^X(\push(\delta_X))}", dashed, from=1-2, to=2-2]
    \end{tikzcd}\]

    Let $\push(\delta_X) = \delta_X$. We need to show that for $x \le_X x'$,
    we have $\ptb_V^X(\delta_X)(x) \le \ptb_V^X(\delta_X)(x')$, which holds because
    $\ptb_V^X(\delta_X)$ is monotone with respect to $\le_X$.


    \item \emph{Horizontal composition}:
    
    Let $R : X \nrightarrow Y$ and $S : Y \nrightarrow Z$.
    We define
    $e_{XZ} = e_{YZ} \circ e_{XY}$ and $p_{XZ} = \ext{p_{XY}}{} \circ p_{YZ}$.
    We define the distinguished perturbations in the representability rules as follows:

    \begin{align*}
        \delta_{R\circ S}^{l,e} &amp;= \pull_R(\delta_S^{l,e}) \cdot \delta_R^{l,e} \\
        \delta_{R\circ S}^{r,e} &amp;= \delta_S^{r,e} \cdot \push_S(\delta_R^{r,e}) \\
        \delta_{R\circ S}^{l,p} &amp;= \delta_R^{l,p} \cdot \pull_{LR}(\delta_S^{l,p}) \\
        \delta_{R\circ S}^{r,p} &amp;= \push_{LS}(\delta_R^{r,p}) \cdot \delta_S^{r,p}
    \end{align*}
    where $\cdot$ denotes composition in the appropriate monoid of perturbations.

    We can then show that the four quasi-representability rules are valid with these definitions.
    \textbf{TODO show one or two of the cases}

    \vspace{3ex}

    We also need to show that the push-pull rules hold of the composition $R \circ S$.
    This follows from the fact that they hold for both $R$ and $S$.
    Specifically, we define

    \begin{align*}
        \push_{R \circ S}(\delta^X) &amp;= \push_S(\push_R(\delta^X)) \\
        \pull_{R \circ S}(\delta^Z) &amp;= \pull_R(\pull_S(\delta^Z)) \\
        \push_{L(R \circ S)}(\delta^X) &amp;= \push_{LS}(\push_{LR}(\delta^X)) \\
        \pull_{L(R \circ S)}(\delta^Z) &amp;= \pull_{LR}(\pull_{LS}(\delta^Z))
    \end{align*}

    Then we can verify that the relevant push-pull inequalities hold using the above definitions.
    \textbf{TODO maybe show one of the cases}

    \item \emph{Identity two-cells}:
    The horizontal identity two-cells have the form

    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWSJdLFsxLDAsIlgiXSxbMSwxLCJZIl0sWzAsMSwiZiIsMl0sWzIsMywiZiJdLFswLDIsIlxcbGVfWCJdLFsxLDMsIlxcbGVfWSIsMl0sWzQsNSwiXFxsdGR5biIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==
    \[\begin{tikzcd}[ampersand replacement=\&amp;]
        X \&amp; X \\
        Y \&amp; Y
        \arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=2-1]
        \arrow[""{name=1, anchor=center, inner sep=0}, "f", from=1-2, to=2-2]
        \arrow["{\le_X}", from=1-1, to=1-2]
        \arrow["{\le_Y}"', from=2-1, to=2-2]
        \arrow["\ltdyn"{description}, draw=none, from=0, to=1]
    \end{tikzcd}\]

    This square commutes because $f$ is monotone with respect to the ordering relation.

    The vertical two-cells have the form

    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsMSwiWCJdLFsxLDAsIlkiXSxbMSwxLCJZIl0sWzAsMiwiUiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzEsMywiUiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzAsMSwiXFxpZF9YIiwyXSxbMiwzLCJcXGlkX1kiXSxbNiw3LCJcXGx0ZHluIiwxLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d
    \[\begin{tikzcd}[ampersand replacement=\&amp;]
        X \&amp; Y \\
        X \&amp; Y
        \arrow["R", "\shortmid"{marking}, no head, from=1-1, to=1-2]
        \arrow["R", "\shortmid"{marking}, no head, from=2-1, to=2-2]
        \arrow[""{name=0, anchor=center, inner sep=0}, "{\id_X}"', from=1-1, to=2-1]
        \arrow[""{name=1, anchor=center, inner sep=0}, "{\id_Y}", from=1-2, to=2-2]
        \arrow["\ltdyn"{description}, draw=none, from=0, to=1],
    \end{tikzcd}\]

    which commutes trivially.

    \item \emph{Composition of two-cells}:
    Two-cells compose vertically and horizontally, which follows from the definition
    definition of a two-cell in this category.

    \textbf{TODO: elaborate}
\end{itemize}

\vspace{3ex}

\textbf{Kleisli internal hom: TODO}

% The arrow =&gt; takes a value type and an algebra and constructs an algebra</pre></body></html>