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\begin{document}

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\section*{thunk-force category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{thunkforce_category}{}\section*{{Thunk-force category}}\label{thunkforce_category}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{thunkable_morphisms}{Thunkable Morphisms}\dotfill \pageref*{thunkable_morphisms} \linebreak
\noindent\hyperlink{relation_to_monads}{Relation to Monads}\dotfill \pageref*{relation_to_monads} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related Concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{thunk-force category} is a category that models [[call-by-value]] programming languages with effects. More commonly, terms in a call-by-value language are modelled as morphisms in the [[Kleisli category]] of a [[strong monad]]. A thunk-force category axiomatizes the Kleisli category \emph{directly} and is also called an \textbf{abstract Kleisli category}\footnote{We prefer the term thunk-force category since it is ambiguous whether [[Kleisli category]] refers to the Kleisli category of a [[monad]] or a [[comonad]].} . The original category can be recovered from this category if it satisfies certain properties.

The name derives from programming terminology. A \textbf{thunk} means a value that represents an unevaluated computation, which can later be \textbf{forced} to be evaluated and perform its [[side effects]]. This is represented by the object $L A$ which could be read as a ``thunk of $A$'' or a ``lazy $A$''. The thunking arrow $\theta_A : A \to L A$ embeds a value as a trivial thunk, and the force $\epsilon_A : L A \to A$ forces the evaluation of its thunked input.

The type constructor $L$ and associated thunk and force don't normally appear as a standalone feature in a call-by-value language, but if the language supports first-class functions, they can be implemented in the usual way as a function of unit input. On the model side, this corresponds to the fact that if a thunk-force category is [[premonoidal category|premonoidal closed]], then $L A$ is equivalent to $I \rightharpoonup A$ and $\theta, \epsilon$ can be defined using this structure.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A thunk-force category \hyperlink{F99}{F\"u{}hrmann 99} consists of

\begin{itemize}%
\item a category $K$
\item a functor $L : K \to K$
\item a [[unnatural transformation|possibly unnatural transformation]] $\theta : Id_K \to L$ pronounced \textbf{thunk}
\item a [[natural transformation]] $\epsilon : L \to Id_K$ pronounced \textbf{force}

\end{itemize}
such that

\begin{itemize}%
\item $(L\theta)\theta = (\theta L)\theta$
\item $\theta L$ is a natural transformation
\item $\epsilon\theta = \id$
\item $(L\epsilon)(\theta L) = \id$

\end{itemize}
The definition can be extended to monoidal closed thunk-force category to model a context and strong monad \hyperlink{F99}{F\"u{}hrmann 99}.

\hypertarget{thunkable_morphisms}{}\subsection*{{Thunkable Morphisms}}\label{thunkable_morphisms}

A morphism $f : A \to B$ represents an effectful program. The presence of the thunk $\theta$ allows us to make the distinction between the ``pure''/``trivially effectful'' programs and those that have non-trivial effects. A morphism $f : A \to B$ in a thunk-force category is \textbf{thunkable} if $( L f )\theta = \theta f$, i.e., $\theta$ is ``natural with respect to $f$''. Then the conditions on the data $(L,\epsilon,\theta)$ in the definition of thunk-force category can be rephrased as

\begin{itemize}%
\item $\theta$ is thunkable
\item $(L,\epsilon,\theta L)$ is a [[comonad]].

\end{itemize}
The presence of non-thunkable morphisms makes $\theta$ fail to be natural, naturality of $\theta$ is exactly the same as saying all morphisms are thunkable. This is the case for the Kleisli category of the identity monad. On the other hand, most monads produce many non-thunkable morphisms. For example in the Kleisli category of the [[maybe monad]], which is equivalent to the category of sets and partial functions, the thunkable morphisms are the total functions. As discussed below, most monads $T$ on a category $C$ used in [[denotational semantics]] satisfy an equalizing requirement which means the thunkable morphisms are in one-to-one correspondence to the morphisms of the original category $C$.

\hypertarget{relation_to_monads}{}\subsection*{{Relation to Monads}}\label{relation_to_monads}

Let $(T,\eta,\mu)$ be a [[monad]] on a category $C$. The [[Kleisli category]] of $T$ is a thunk-force category. Let $F \dashv G$ denote the adjunction between $C$ and the Kleisli category $C_T$. Define $L = FG$ and $\epsilon$ the counit of the comonad. To define $\theta$, take $F\eta : F \to FT$ and, using the fact that $F$ is an [[identity-on-objects functor]] (or at least bijective on objects), treat it as an [[unnatural transformation]] from $Id_{C_T}\to L$. Then it follows immediately by naturality that every morphism in the image of $F$ is thunkable.

More explicitly, using the ``Kleisli arrow'' definition of the Kleisli category, $L$ is just $T$, $\epsilon_A : T A \to T A$ is the identity, and $\theta_A = \eta_{T A} \eta_{A} : A \to T^2 A$.

In the opposite direction, starting with a thunk-force category $K$, we can define the (non-full) subcategory of thunkable morphisms $G_\theta K$. Since $\theta L$ is natural, the image of $L$ lands in this subcategory, and is right adjoint to the inclusion. Then there is a monad $L_\theta$ induced on $G_{\theta}K$ by this adjunction. Furthermore, the thunk-force category generated by the Kleisli category of the monad is equivalent to $K$.

These constructions exhibit the category of thunk-force categories as a [[reflective subcategory]] of the category of monads. The subcategory can be characterized as the monads satisfying the ``equalizing requirement'', that the unit $\eta$ of the monad is an equalizer in the following diagram:

\begin{displaymath}
1_{C}\underset{\quad \eta \quad}{\to}T\underoverset{\quad \eta T \quad}{T \eta}{\rightrightarrows}T^2
\end{displaymath}
Which is in practice often satisfied for monads used in [[denotational semantics]]. The definition of the category of morphism of thunk-force categories and proof of this theorem are in \hyperlink{F99}{F\"u{}hrmann 99 section 5}.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

Since the thunk $L$ is a [[comonad]], the Kleisli category of the comonad is a model of a [[call-by-name]] language. This is a semantic counterpart of the ``thunking translation'' of call-by-name into call-by-value (described for example in \hyperlink{HD97}{Hatcliff-Danvy 97}.

\hypertarget{related_concepts}{}\subsection*{{Related Concepts}}\label{related_concepts}

\begin{itemize}%
\item The dual concept for [[call-by-name]] is a [[runnable monad]].
\item A [[duploid]] is an analogous structure for effectful languages with both [[positive type|positive types]] and [[negative type|negative types]].

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Thunk-force categories were introduced (under the name ``abstract Kleisli categories'' in

\begin{itemize}%
\item Carsten F\"u{}hrmann, ``Direct Models of the Computational Lambda-calculus'' Electronic Notes in Theoretical Computer Science 1999


\item John Hatcliff and Olivier Danvy, ``Thunks and the $\lambda$-Calculus'', Journal of Functional Programming 1997



\end{itemize}


\end{document}