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\begin{document}

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\section*{modality}

\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{IdeaInPhilosophy}{In philosophy}\dotfill \pageref*{IdeaInPhilosophy} \linebreak
\noindent\hyperlink{InFormalLogic}{In formal logic}\dotfill \pageref*{InFormalLogic} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{Notation}{Notation}\dotfill \pageref*{Notation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{in_formal_logic_2}{In formal logic}\dotfill \pageref*{in_formal_logic_2} \linebreak
\noindent\hyperlink{in_philosopy}{In philosopy}\dotfill \pageref*{in_philosopy} \linebreak


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

A \emph{modality} in [[philosophy]] and formally in [[formal logic]]/[[type theory]] expresses a certain \emph{mode} (or ``moment'' as in \hyperlink{Hegel12}{Hegel 12}) of [[being]].

\hypertarget{IdeaInPhilosophy}{}\subsubsection*{{In philosophy}}\label{IdeaInPhilosophy}

According to (\hyperlink{Kant1900}{Kant 1900}) (see \href{http://de.wikipedia.org/wiki/Kategorie_%28Philosophie%29}{WP}) the four ``[[category (philosophy)|categories]]'' are

\begin{enumerate}%
\item Quantity


\item Quality


\item Relation


\item Modality



\end{enumerate}
and the modalities contain the three [[unity of opposites|pairs of opposites]]

\begin{itemize}%
\item [[possibility]] - impossibility


\item [[being]] - [[nothing]]


\item [[necessity]] - Zuf\"a{}lligkeit



\end{itemize}
\hypertarget{InFormalLogic}{}\subsubsection*{{In formal logic}}\label{InFormalLogic}

\hypertarget{general}{}\paragraph*{{General}}\label{general}

In [[formal logic]] and [[type theory]] modalities are formalized by \emph{modal operator} or \emph{[[closure operator]]} $\sharp$, that send [[propositions]]/[[types]] $X$ to new propositions/types $\sharp X$, satisfying some properties.

Adding such modalities to [[propositional logic]] or similar produces what is called [[modal logic]]. Here operators that are meant to formalize [[necessity]] and [[possibility]] ([[S4 modal logic]]) are maybe most famous. Adding modalities more generally to [[type theory]] yields \emph{[[modal type theory]]}. See there for more details.

The [[categorical semantics]] of these modalities is that $\sharp$ is interpreted an [[idempotent monad]]/[[comonad]] on the [[category of contexts]].

This has a refinement to [[homotopy type theory]], where the [[categorical semantics]] of a \emph{higher modality} or \emph{homotopy modality} as an idempotent [[(infinity,1)-monad]] (\hyperlink{Shulman}{Shulman 12}, \hyperlink{RSS}{Rijke, Shulman, Spitters} ).

\hypertarget{Notation}{}\paragraph*{{Notation}}\label{Notation}

Typical notation (e.g. \hyperlink{SEP}{SEP}, \hyperlink{Reyes91}{Reyes 91}, but not \hyperlink{Hermida10}{Hermida 10}) is as follows:

\begin{itemize}%
\item a co-modality represented by an [[idempotent comonad]] is typically denoted by $\Box$, following the traditional example of \emph{[[necessity]]} in [[modal logic]];


\item a modality represented by an [[idempotent monad]] is typically denoted by $\lozenge$ or (less often) by $\bigcirc$, following the traditional example of \emph{[[possibility]]} in [[modal logic]].



\end{itemize}
When [[adjunctions]] between modalities matter ([[adjoint modalities]]), then some authors (\hyperlink{Reyes91}{Reyes 91, p. 367} \hyperlink{ReyesEtAl}{RRZ 04, p. 116}, \hyperlink{Hermida10}{Hermida 10, p.11}) use $\lozenge$ for a [[left adjoint]] of a $\Box$. That leaves $\bigcirc$ as the natural choice of notation for a [[right adjoint]] (if any) of a $\Box$-modality.

This way for instance for [[cohesion]] with [[shape modality]] $\dashv$ [[flat modality|flat comodality]] $\dashv$ [[sharp modality]] the generic notation would be:

\begin{displaymath}
\itexarray{
     monad && comonad && monad
     \\
     \lozenge &\dashv& \Box &\dashv& \bigcirc
     \\
     \\
     shape && flat && sharp
     \\
     &#643; &\dashv& \flat &\dashv& \sharp 
  }
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[necessity]] $\dashv$ [[possibility]]


\item [[local modality]], [[Lawvere-Tierney topology]]


\item [[double negation modality]]


\item [[n-truncation modality]]


\item [[unit type]] modality, [[empty type]] co-modality ([[nothing]] $\dashv$ [[being]])


\item [[exponential modality]]


\item [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]


\item [[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]


\item [[affine modality]]


\item [[graded modality]]



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

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


\item [[de dicto and de re]]


\item [[monad (in computer science)]]


\item [[modal type]], [[anti-modal type]]


\item [[adjoint modality]], [[adjoint logic]]


\item [[Galois connection]]



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

\hypertarget{in_formal_logic_2}{}\subsubsection*{{In formal logic}}\label{in_formal_logic_2}

Discussion in [[formal logic]] and [[homotopy type theory]] ([[modal type theory]]):

\begin{itemize}%
\item [[Mike Shulman]], \emph{Higher modalities}, talk at [[UF-IAS-2012]], October 2012 (\href{http://uf-ias-2012.wikispaces.com/file/view/modalitt.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{http://homotopytypetheory.org/2012/11/19/all-modalities-are-hits/}{All modalities are Higher Inductive Types}}


\item [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], \emph{Modalities in homotopy type theory} \href{https://arxiv.org/abs/1706.07526}{arXiv}



\end{itemize}
\hypertarget{in_philosopy}{}\subsubsection*{{In philosopy}}\label{in_philosopy}

\begin{itemize}%
\item [[Georg Hegel]], \emph{[[Science of Logic]]}, 1812

\end{itemize}
\begin{itemize}%
\item [[Kant]], AA III, 93-- KrV B 106

\end{itemize}
\begin{itemize}%
\item German Wikipedia, \emph{\href{https://de.wikipedia.org/?curid=3382094}{Modalit\"a{}t (Philosophie)}}


\item Stanford Encyclopedia of Philosophy, \emph{\href{http://plato.stanford.edu/entries/logic-modal/}{Modal Logic}}


\item [[Gonzalo Reyes]], \emph{A topos-theoretic approach to reference and modality}, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (\href{http://projecteuclid.org/euclid.ndjfl/1093635834}{Euclid})


\item Reyes/Reyes/Zolfaghari, \emph{Generic Figures and Their Glueings} 2004, Polimetrica


\item [[Claudio Hermida]], section 3.3. of \emph{A categorical outlook on relational modalities and simulations}, 2010 (\href{http://maggie.cs.queensu.ca/chermida/papers/sat-sim-IandC.pdf}{pdf})



\end{itemize}
[[!redirects modality]] [[!redirects modalities]] [[!redirects modal operator]] [[!redirects modal operators]]

[[!redirects comodality]] [[!redirects comodalities]] [[!redirects comodal operator]] [[!redirects comodal operators]]

[[!redirects co-modality]] [[!redirects co-modalities]] [[!redirects co-modal operator]] [[!redirects co-modal operators]]



\end{document}