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\begin{document}

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\section*{maybe monad}

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{kleisli_category}{Kleisli category}\dotfill \pageref*{kleisli_category} \linebreak
\noindent\hyperlink{EMCategoryAndRelationToPointedObjects}{EM-category and Relation to pointed objects}\dotfill \pageref*{EMCategoryAndRelationToPointedObjects} \linebreak
\noindent\hyperlink{relation_to_natural_number_objects}{Relation to natural number objects}\dotfill \pageref*{relation_to_natural_number_objects} \linebreak
\noindent\hyperlink{on_the_augmented_simplex_category}{On the augmented simplex category}\dotfill \pageref*{on_the_augmented_simplex_category} \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}

What is called the \emph{maybe monad} is a simple [[monad (in computer science)]] which is used to implement ``exceptions'' indicating the failure of a computation in terms of [[functional programming]].

On the type system the [[maybe monad]] is the operation $X \mapsto X \coprod \ast$.

The idea here is that a function $X \longrightarrow Y$ in its [[Kleisli category]] is in the original category a function of the form $X \longrightarrow Y \coprod \ast$ so either returns indeed a value in $Y$ or else returns the unique element of the [[unit type]]/[[terminal object]] $\ast$ -- it is a \emph{[[partial function]]}. The latter case is naturally interpreted as ``no value returned'', hence as indicating a ``failure in computation''.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{quote}%
under construction


\end{quote}
\hypertarget{kleisli_category}{}\subsubsection*{{Kleisli category}}\label{kleisli_category}

The [[Kleisli category]] of the maybe monad on $Set$ is the category whose objects are sets, and whose morphisms are [[partial functions]].

This observation generalizes as follows: if $\mathbf{C}$ is an [[extensive category]] and has a [[terminal object]] $\ast$, then a morphism $g: X \to Y \coprod \ast$ as an object of the [[overcategory]] $\mathbf(C)/(Y \coprod \ast)$ determines (uniquely up to canonical isomorphism) an object

\begin{displaymath}
(f: X_1 \to Y, !: X_2 \to \ast) \in \mathbf{C}/Y \times \mathbf{C}/\ast
\end{displaymath}
such that $g = f \coprod !$, in other words a partial morphism $f: X \rightharpoonup Y$ whose domain of definition is a subobject $X_1 \hookrightarrow X$ with complement $X_2 \hookrightarrow X$. In brief, maps in the Kleisli category are partial maps with complemented domain.

In particular, in the case of a [[Boolean topos]], the Kleisli category is the category of objects and partial maps; see also [[partial map classifier]].

\hypertarget{EMCategoryAndRelationToPointedObjects}{}\subsubsection*{{EM-category and Relation to pointed objects}}\label{EMCategoryAndRelationToPointedObjects}

The [[algebra over a monad|algebras]] over the maybe monad are [[pointed objects]].

Moreover, assuming $\mathbf{C}$ has [[finite products]] and an appropriate form of [[distributive category|distributivity]] (which obtains if for example $\mathbf{C}$ is [[extensive category|lextensive]]), the maybe monad on $\mathbf{C}$ is a [[monoidal monad]] on the [[cartesian monoidal category]] $\mathbf{C}$. It follows (by the discussion at \emph{[[commutative monad]]}, see also (\hyperlink{Seal12}{Seal 12})) that its [[Eilenberg-Moore category]] of algebras canonically inherits the structure of a [[monoidal category]], at least under the mild assumption that it has [[reflexive coequalizers]]. Note that the maybe monad $T$ preserves reflexive coequalizers, so the monadic functor [[created limit|creates]] reflexive coequalizers if the base category has them; in this abstract setting the monoidal product on algebras $(X, \alpha: T X \to X)$, $(Y, \beta: T Y \to Y)$ is given explicitly as the [[coequalizer]] of $T(\alpha \times \beta): T(T X \times T Y) \to T(X \times Y)$ and

\begin{displaymath}
T(T X \times T Y) \stackrel{T(\phi_{X, Y})}{\to} T T(X \times Y) \stackrel{\mu}{\to} T(X \times Y)
\end{displaymath}
where $\phi$ is one of the structural constraints on the monoidal monad $T$ and $\mu$ is the multiplication on $T$. One finds that this coequalizer yields the usual [[smash product]] of pointed objects.

\begin{remark}
\label{}\hypertarget{}{}
The smash product as the correct monoidal product can also be deduced in a perhaps more perspicuous manner if we assume more of the base category: that it is [[cartesian closed category|cartesian closed]], [[finite limit|finitely complete]], and finitely cocomplete. In that case we construct the [[internal hom]] of $T$-algebras, i.e., the internal hom of pointed objects $(Y, \beta: T Y \to Y)$ and $(Z, \gamma: T Z \to Z)$ directly as an [[equalizer]] of maps

\begin{displaymath}
\itexarray{
Z^Y & \to & T Z^{T Y} \\ 
 & \mathllap{Z^\beta} \searrow & \downarrow \mathrlap{\gamma^{T Y}} \\
 & & Z^{T Y}
}
\end{displaymath}
where the top arrow expresses [[enriched functor|enriched functoriality]] of $T$ (which in turn is closely related to the [[strong functor|strength]] on $T$). The success of this is guaranteed by the [[commutative monad|commutativity]] of the monad (which here takes a particularly simple form, being given by the commutative \emph{[[monoid]]} $\ast$ with respect to coproduct $\coprod$). Then, by taking the monoidal product that is [[adjunct|adjoint]] to the [[internal hom]], one is led to the [[smash product]] $(X \wedge Y)_\ast$ all the same: that is, one can read off the smash product from the fact that pointed maps $X_\ast \to \hom_\ast(Y_\ast, Z_\ast)$ should correspond to pointed maps $(X \wedge Y)_\ast \to Z_\ast$.

\end{remark}
\hypertarget{relation_to_natural_number_objects}{}\subsubsection*{{Relation to natural number objects}}\label{relation_to_natural_number_objects}

Regarding just the underlying [[endofunctor]] of the maybe monad, its [[initial algebra over an endofunctor]] is a [[natural numbers object]].

\hypertarget{on_the_augmented_simplex_category}{}\subsubsection*{{On the augmented simplex category}}\label{on_the_augmented_simplex_category}

We may view the [[simplex category|augmented simplex category]] as the subcategory of $\Set$ whose objects are the finite [[von Neumann ordinals]] and whose morphisms are the monotone functions between them. Then the maybe monad on $\Set$ restricts to $\Delta_a$ to give the monad that sends the object $\mathbf{n}$ to $\mathbf{n+1}$ and the morphism $f:\mathbf{n}\to\mathbf{m}$ to the morphism $T(f):\mathbf{n+1}\to\mathbf{m+1}$ defined by

\begin{displaymath}
T(f)(k) = \begin{cases}
f(k) & k \lt n\\
m & k = n
\end{cases}
\end{displaymath}
In fact, $\Delta_a$ is freely generated by this structure and $\mathbf{0}$ in the sense that its objects are given by $\mathbf{n}=T^n\mathbf{0}$, the face maps are given by $\delta_i^n=T^{n-i-1}\eta_\mathbf{i}$, the degeneracy maps are given by $\sigma_i^n=T^{n-i-1}\mu_\mathbf{i}$, and the [[simplicial identities]] are precisely the monad axioms. Another way to put this is that $(\Delta_a,T,\mathbf{0})$ is the initial object of the $2$-category whose objects are categories equipped with a monad and an object.

This means that if C is some other category equipped with a \textbf{co}monad and an object then we get a canonical functor $\Delta_a^\mathrm{op}\to C$ and hence an augmented [[simplicial object]] in $C$. In particular when $C$ is the [[Eilenberg-Moore category| category of algebras]] of a monad on $D$ we get a simplicial object for each algebra, whose underlying simplicial object in $D$ is the [[bar construction]].

The comonad $T^\mathrm{op}$ on $\Delta_a^\mathrm{op}$ induces the [[decalage\#DecalageComonad|Décalage comonad]].

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

\begin{itemize}%
\item [[state monad]]


\item [[continuation monad]]


\item [[successor monad]]



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

\begin{itemize}%
\item Gavin J. Seal, \emph{Tensors, monads and actions} (\href{http://arxiv.org/abs/1205.0101}{arXiv:1205.0101})

\end{itemize}
Around (0.4.24.2) in

\begin{itemize}%
\item [[Nikolai Durov]], \emph{New Approach to Arakelov Geometry} (\href{http://arxiv.org/abs/0704.2030}{arXiv:0704.2030})

\end{itemize}
the algebraic structure of the would be ``[[field with one element]]'' is regarded as being the maybe monad, hence [[modules]] over $\mathbb{F}_1$ are defined to be [[algebra over a monad|monad-algebras]] over the maybe monad, hence [[pointed sets]].

[[!redirects maybe monads]]



\end{document}