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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{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{modal_term_calculi}{Modal term calculi}\dotfill \pageref*{modal_term_calculi} \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}

In full [[linear logic]]/[[linear type theory]] there is assumed a (comonadic) [[modality]] denoted ``!'' and called the \emph{exponential modality}, whose role is, roughly, to give linear types also a non-linear interpretation. This is also called the ``of course''-modality or the \emph{storage modality}, and sometimes the ``bang''-operation.

In classical linear logic (meaning with involutive de Morgan duality), the de Morgan dual of ``!'' is denoted ``?'' and called the ``why not''-modality.

In [[categorical semantics]] of linear type theory the !-modality typically appears as a kind of [[Fock space]] construction. If one views [[linear logic]] as [[quantum logic]] (as discussed there), then this means that the !-modality produces [[free field theory|free]] [[second quantization]].

\hypertarget{categorical_semantics}{}\subsection*{{Categorical semantics}}\label{categorical_semantics}

Everyone agrees that ! should be a comonad (and ? should be a monad), but there are different ways to proceed from there. The goal is to capture the syntactic rules allowing assumptions of the form $!A$ to be duplicated and discarded. The original definition from \hyperlink{Seely89}{Seely} was:

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be an $\ast$-[[star-autonomous category|autonomous]] category with [[cartesian products]]. A Seely !-modality on $C$ is a comonad that is a [[strong monoidal functor]] from the [[cartesian monoidal category|cartesian monoidal structure]] to the $\ast$-autonomous monoidal structure, i.e. we have $!(A\times B)\cong !A \otimes !B$ coherently. (There is also a [[coherence]] [[axiom]] that should be imposed; see \hyperlink{Mellies09}{Mellies, section 7.3}.)

\end{defn}
(Note that in linear logic, the cartesian monoidal structure $\times$ is sometimes denoted by $\&$.) This implies that the [[Kleisli category of a comonad|Kleisli category]] of ! is a [[cartesian closed category]], which is a categorical version of the translation of intuitionistic logic into linear logic.

Of course, the above definition depends on the existence of the cartesian product, and relies on the self-duality of an $\ast$-autonomous category to derive the rules for ? from the rules for !. A different definition that doesn't require the existence of $\times$ was given by \hyperlink{BBPH92}{Benton, Bierman, de Paiva, and Hyland}:

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[closed symmetric monoidal category]]; a !-modality on $C$ is a [[lax monoidal functor|lax monoidal]] comonad such that every !-coalgebra naturally carries the structure of a [[comonoid object]] in the category of coalgebras, such that coalgebra maps are comonoid maps.

\end{defn}
This definition implies that the category of all !-coalgebras (not just the free ones, i.e. its Kleisli category) is cartesian closed. Note that for a comonad on a [[poset]], every coalgebra is free; thus the world of pure propositional ``logic'' doesn't tell us whether to consider the Kleisli category or the Eilenberg-Moore category for the translation. A more even-handed approach is the following (see \hyperlink{Mellies09}{Mellies}):

\begin{defn}
\label{}\hypertarget{}{}
A \emph{linear-nonlinear adjunction} is a [[monoidal adjunction]] $F : M \rightleftarrows L : G$ in which $L$ is symmetric monoidal and $M$ is cartesian monoidal. The induced !-modality is the comonad $F G$ on $L$.

\end{defn}
This includes both of the previous definitions where $M$ is taken respectively to be the Kleisli category or the Eilenberg-Moore category of !.

On the other hand, the last two definitions are given only for ``intuitionistic'' linear logic, though in the $\ast$-autonomous case one could derive a ? from the ! by de Morgan duality. A definition not requiring the de Morgan duality and describing ! and ? together was given by \hyperlink{BCS96}{Blute, Cockett, and Seely}:

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[linearly distributive category]] with tensor product $\otimes$ and cotensor product $\parr$. A (!,?)-modality on $C$ consists of:

\begin{enumerate}%
\item a $\otimes$-monoidal comonad ! and a $\parr$-comonoidal monad ?
\item ? is a !-strong monad, and ! is a ∞-strong comonad
\item all free !-coalgebras are naturally commutative $\otimes$-comonoids, and all free ∞-algebras are naturally commutative $\parr$-monoids.

\end{enumerate}
\end{defn}
Here a functor $F$ is [[strong functor|strong]] with respect to a lax monoidal functor $G$ if there is a natural transformation $F A \otimes G B \to F(A\otimes G B)$ satisfying some natural axioms, and we similarly require compatibility of the monad and comonad structure transformations.

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

$\backslash$begin\{theorem\} Suppose $C$ is a [[cartesian monoidal category]] with finite limits and colimits, and $\bot\in C$ is an object. Then the [[Chu construction]] $Chu(C,\bot)$, which is $\ast$-autonomous with finite limits and colimits, admits a Seely !-modality, namely the [[idempotent comonad]] induced by the [[coreflective subcategory|coreflective]] embedding of $C$. $\backslash$end\{theorem\} $\backslash$begin\{proof\} The embedding of any $C$ in its Chu construction as $A \mapsto (A, [A,\bot], ev)$ is coreflective: the coreflection of $(B^+, B^-, e_B)$ is $(B^+, [B^+,\bot], ev)$. Moreover, this subcategory is closed under the tensor product of $Chu(C,\bot)$, i.e. the embedding $C\hookrightarrow Chu(C,\bot)$ is strong monoidal. The Seely condition therefore means more concretely that the coreflection takes the cartesian product in $Chu(C,\bot)$ to the tensor product of $C$. But when $C$ is cartesian monoidal, its tensor product is also the cartesian product, so this follows from the fact that the coreflection is a [[right adjoint]] and hence preserves products. $\backslash$end\{proof\}

\hypertarget{modal_term_calculi}{}\subsection*{{Modal term calculi}}\label{modal_term_calculi}

Girard's original presentation of linear logic involved rules that explicitly assumed the presence of $!$ on hypotheses or on entire contexts, such as [[weakening rule|weakening]] and [[contraction rule|contraction]]:

\begin{displaymath}
\frac{\Gamma \vdash B}{\Gamma, !A\vdash B} \qquad \frac{\Gamma,!A,!A \vdash B}{\Gamma, !A\vdash B}
\end{displaymath}
and ``promotion'':

\begin{displaymath}
\frac{!\Gamma \vdash A}{!\Gamma\vdash !A}
\end{displaymath}
If this is translated into a [[natural deduction]] style term calculus, the resulting rules are more complicated than those of most type formers. This can be avoided using [[adjoint type theory]] with two context zones, one ``nonlinear'' one where contraction and weakening are permitted (and [[admissible rule|admissible]]) and one ``linear'' one where they are not, with $!$ as a modality relating the two zones.

Such a ``modal'' presentation of linear logic was first introduced by Girard in his work on [[LU]] and then developed by a number of other people such as Plotkin, Wadler, Benton, and Barber. See the references for details.

This presentation also generalizes naturally to [[dependent linear type theory]], with the nonlinear type theory being dependent, and the linear types depending on the nonlinear ones but nothing depending on linear types. In this context, the $!$-modality decomposes into ``context extension'' and a ``dependent sum''.

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

\begin{itemize}%
\item [[exponential map]]


\item [[multiplicative conjunction]]


\item [[linear implication]]



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

The semantics of ! as a comonad is discussed in:

\begin{itemize}%
\item [[R. A. G. Seely]], \emph{Linear logic, $\ast$-autonomous categories and cofree coalgebras}, \emph{Contemporary Mathematics} 92, 1989. ([[SeelyLinearLogic.pdf:file]], \href{http://www.math.mcgill.ca/rags/nets/llsac.ps.gz}{ps.gz})


\item [[Nick Benton]], Gavin Bierman, [[Valeria de Paiva]], [[Martin Hyland]], \emph{Linear lambda-Calculus and Categorical Models Revisited} (1992), \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.6958}{citeseer}


\item R. F. Blute , J. R. B. Cockett and R. A. G. Seely. \emph{! and ? -- Storage as tensorial strength} \href{https://doi.org/10.1017/S0960129500001055}{doi}, \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.7317&rep=rep1&type=pdf}{pdf}


\item [[Paul-André Melliès]], \emph{Categorical semantics of linear logic}, 2009. \href{https://www.irif.fr/~mellies/papers/panorama.pdf}{pdf}


\item [[Martin Hyland]] and [[Andreas Schalk]], \emph{Glueing and orthogonality for models of linear logic}, \href{http://www.cs.man.ac.uk/~schalk/publ/gomll.pdf}{pdf}



\end{itemize}
The relation to Fock space is discussed in:

\begin{itemize}%
\item [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], \emph{Fock Space: A Model of Linear Exponential Types} (1994) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825}{web}, [[BPSLinear.pdf:file]])


\item [[Marcelo Fiore]], \emph{Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic}, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 (\href{http://www.cl.cam.ac.uk/~mpf23/papers/Types/diff.pdf}{pdf})


\item [[Jamie Vicary]], \emph{A categorical framework for the quantum harmonic oscillator}, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 (\href{http://arxiv.org/abs/0706.0711}{arXiv:0706.0711})

(in the context of [[finite quantum mechanics in terms of dagger-compact categories]])



\end{itemize}
The interpretation of $\Omega^\infty \Sigma^\infty_+$ as an exponential in the context of [[Goodwillie calculus]] is due to

\begin{itemize}%
\item [[Gregory Arone]], Marja Kankaanrinta, \emph{The Goodwillie tower of the identity is a logarithm}, 1995 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306}{web})

\end{itemize}
based on

\begin{itemize}%
\item [[Gregory Arone]], [[Mark Mahowald]], \emph{The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres}, 1998 (\href{http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf}{pdf})

\end{itemize}
The modal approach to a term calculus for the $!$-modality can be found in:

\begin{itemize}%
\item [[Jean-Yves Girard]]. \emph{On the unity of logic.} Annals of Pure and Applied Logic, 59:201-217, 1993.


\item G. Plotkin. \emph{Type theory and recursion.} In Proceedings of the Eigth Symposium of Logic in Computer Science, Montreal , page 374. IEEE Computer Society Press, 1993.


\item N. Benton. \emph{A mixed linear and non-linear logic; proofs, terms and models.} In Proceedings of Computer Science Logic `94, number 933 in LNCS. Verlag, June 1995.


\item Philip Wadler. \emph{A syntax for linear logic.} In Ninth International Coference on the Mathematical Foundations of Programming Semantics , volume 802 of LNCS . Springer Verlag, April 1993


\item Andrew Barber, \emph{Dual Intuitionistic Linear Logic}, Technical Report ECS-LFCS-96-347, University of Edinburgh, Edinburgh (1996), \href{http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/}{web}



\end{itemize}
[[!redirects of course]] [[!redirects !]] [[!redirects why not]] [[!redirects ?]] [[!redirects exponential conjunction]] [[!redirects exponential modality]] [[!redirects storage modality]] [[!redirects exponential modalities]] [[!redirects storage modalities]]



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