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\begin{document}

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\section*{modal type}

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

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

[[!include type theory - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In [[modal logic]] a [[modality]] expresses a ``way of being true'' and a modal proposition (or [[stable proposition]]) is a [[proposition]] which is indeed [[true]] in the given way (for instance being [[necessity|necessarily]] true or [[possibility|possibly]] true, in [[S4 modal logic]] ).

As one passes from [[logic]] to ([[homotopy type theory|homotopy]]) [[type theory]] and hence from [[modal logic]] to [[modal type theory]], then being [[true]] is just the lowest stage of a hierarchy of [[truncated object of an (infinity,1)-category|truncated]] types (``[[h-level]]''). Hence for a general [[type]]/[[homotopy type]] then a [[modality]] expresses just a ``way of being''.

For instance if $C$ is a discrete finite [[type]] ([[h-set]]) thought of as a type of ``colors'' and one [[term]] $g \colon C$ of it is thought of as the color ``green'', then the $C$-[[dependent type|dependent types]] may be thought of as colored types; and so there is a [[modal operator]] whose modal types are those which are unicolored in green. Hence here the ``way of being'' expressed by the modality is ``being green''.

More practical examples arise for instance in [[cohesive homotopy type theory]], where for instance the [[flat modality]] expresses the ``way of being [[discrete object|geometrically discrete]]'' and the [[sharp modality]] expresses the ``way of being [[codiscrete object|codiscrete]]''.

If one also regards non-idempotent (co-)monads as [[modal operators]] then the ``way of being'' expressed by them may involve [[structure]] and not just [[property]]. For instance the modal types of the [[maybe monad]] are the [[pointed objects]] and hence the maybe modality expresses the ``way of being pointed''.

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

Given a [[modal type theory]], hence [[type theory]] equipped with a [[closure operator]] [[modality]] $\Diamond$ ([[idempotent monad|idempotent]] [[monad|monadic]]) or $\Box$ ([[idempotent comonad|idempotent]] [[comonad|comonadic]]), the a [[type]] $X$ is \emph{modal} with respect to $\Diamond$/$\Box$ if

\begin{itemize}%
\item the [[unit of an adjunction|unit]] $\eta \colon X \to \Diamond X$


\item or the counit $\epsilon \colon \Box X \to X$



\end{itemize}
is an [[equivalence]].

The collection of modal types forms the \emph{closure} of the given closure operator.

Under [[propositions as types]] a [[proposition]] that is modal is also called a \emph{[[stable proposition]]}.

By the discussion at \emph{\href{idempotent+monad#AlgebrasForAnIdempotentMonad}{idempotent monad -- Properties -- Eilenberg-Moore category of algebra}} the modal types over an idempotent (co-)modality are precisely the types which are (co-)[[algebras over a monad|algebras]] over the given (co-)monad. Hence more generally it makes sense to regard a not-necessarily idempotent (co-)monad as a modal operator and regard its [[algebra over a monad|algebras]] as the corresponding modal types.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

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

\begin{itemize}%
\item [[modality]]


\item [[reflective subcategory]]


\item [[reflective sub-(∞,1)-category]]


\item [[modal type theory]]


\item [[reflective subuniverse]]



\end{itemize}
[[!redirects modal types]]

[[!redirects modal object]] [[!redirects modal objects]]



\end{document}