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\section*{Geometry of Interaction}

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

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

[[!include type theory - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToQuantumOperations}{Relation to superoperators and quantum operations}\dotfill \pageref*{RelationToQuantumOperations} \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 has been called \emph{Geometry of Interaction} (\hyperlink{Girard89}{Girard 89}) is a kind of [[semantics]] for [[linear logic]]/[[linear type theory]] that is however different in method from the usual [[categorical semantics]] in [[monoidal categories]]. Instead of interpreting a [[proof]] of a linear entailment $A\vdash B$ as a [[morphism]] between [[objects]] $A$ and $B$ in a [[monoidal category]] as in [[categorical semantics]], the \emph{Geometry of Interaction} interprets it as an [[endomorphism]] on the object $A\multimap B$. This has been named \emph{operational semantics} to contrast with the traditional \emph{denotational semantics}.

That also the ``operational semantics'' of GoI has an interpretation in [[category theory]], though, namely in [[compact closed categories]] induced from [[traced monoidal categories]] was first suggested in (\hyperlink{JoyalStreetVerity96}{Joyal-Street-Verity 96}) and then developed in (\hyperlink{Haghverdi00}{Haghverdi 00}, \hyperlink{AbramskyHaghverdiScott02}{Abramsky-Haghverdi-Scott 02}, \hyperlink{HaghverdiScott05}{Haghverdi-Scott 05}). See (\hyperlink{Shirahata}{Shirahata}) for a good review.

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

\hypertarget{RelationToQuantumOperations}{}\subsubsection*{{Relation to superoperators and quantum operations}}\label{RelationToQuantumOperations}

We discuss a relation of the GoI to [[superoperators]]/[[quantum operations]] in [[quantum physics]].

According to (\hyperlink{AbramskyHaghverdiScott02}{Abramsky-Haghverdi-Scott 02, remark 5.8}, \hyperlink{Haghverdi00b}{Haghverdi 00b, section 6}) all the standard and intended interpretations of the GoI take place in those [[compact closed categories]] $Int(\mathcal{C})$ free on a [[traced monoidal category]] $\mathcal{C}$ (as discussed there). In these references therefore the notation $Int(-)$ from (\hyperlink{JoyalStreetVerity96}{Joyal-Street-Verity 96}) is changed to $\mathcal{G}(-)$, for ``Geometry of Interaction construction'' (see \hyperlink{AbramskyHaghverdiScott02}{Abramsky-Haghverdi-Scott 02, p. 11}).

The objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism in $\mathcal{C}$ of the form

\begin{displaymath}
\itexarray{
    A^+\otimes B^- 
    \\
    \downarrow
    \\ 
    A^- \otimes B^+
  }
\end{displaymath}
in $\mathcal{C}$, and [[composition]] of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by [[trace|tracing out]] $B^+$ and $B^-$ in the evident way.

We observe now that subcategories of such $Int(\mathcal{C})$ are famous in [[quantum physics]] as ``categories of [[superoperators]]'' or ``categories of [[quantum operations]]'', formalizing [[linear maps]] on spaces of [[operators]] on a [[Hilbert spaces]] that takes [[density matrices]] (hence mixed [[quantum states]]) to density matrices (``[[completely positive maps]]'').

First notice that if the [[traced monoidal category]] structure on $\mathcal{C}$ happens to be that induced by a [[compact closed category]] structure, then by compact closure the morphisms $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ above are equivalently morphisms in $\mathcal{C}$ of the form

\begin{displaymath}
(A^-)^\ast \otimes A^+ \longrightarrow (B^-)^\ast \otimes B^+
\end{displaymath}
and if moreover one concentrates on the case where $A \coloneqq A^- \simeq A^+$ etc. then these are morphisms of the form

\begin{displaymath}
End(A) \longrightarrow End(B)
  \,.
\end{displaymath}
Under this identification the somewhat curious-looking composition in $Int(\mathcal{C})$ given by tracing in $\mathcal{C}$ becomes just plain composition in $\mathcal{C}$.

Subcategories (non-full) of $Int(\mathcal{C})$ consisting of morphisms of this form in some ambient $\mathcal{C}$ have been considered in (\hyperlink{Selinger05}{Selinger 05}), there denoted $CMP(\mathcal{C})$, and shown to formalize the concept of [[quantum operations]] in [[quantum physics]]. For a quick review of Selinger's construction see for instance (\hyperlink{CoeckeHeunen11}{Coecke-Heunen 11, section 2}). Comments related to the relation between GoI and such constructions appear in (\hyperlink{AbramskyCoecke02}{Abramsky-Coecke 02}).

Now the $CPM(-)$ construction requires and assumes that $\mathcal{C}$ has all [[dual objects]] (which in the standard model means to restrict to [[Hilbert spaces]] of [[quantum states]] with are of [[finite]] [[dimension]]). To generalize away from this requirement a more general definition of quantum operations in $\mathcal{C}$ is given in (\hyperlink{CoeckeHeunen11}{Coecke-Heunen 11, def. 1}), there denoted $CP(\mathcal{C})$. Direct inspection shows that $CP(\mathcal{C})$ is still a (non-full) subcategory of $Int(\mathcal{C})$.

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

\begin{itemize}%
\item [[transcendental syntax]]

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

The original references are

\begin{itemize}%
\item [[Jean-Yves Girard]]. Towards a geometry of interaction. In Categories in Computer Science and Logic, pages 69 -- 108, Providence, 1989. American Mathematical Society. Proceedings of Symposia in Pure Mathematics n92.

\end{itemize}
\begin{itemize}%
\item [[Jean-Yves Girard]]. Geometry of interaction I: interpretation of system F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium `88, pages 221 -- 260, Amsterdam, 1989. North-Holland.


\item [[Jean-Yves Girard]]. Geometry of interaction II : deadlock-free algorithms. In Martin-L\"o{}f and Mints, editors, Proceedings of COLOG 88, volume 417 of Lecture Notes in Computer Science, pages 76 -- 93, Heidelberg, 1990. Springer-Verlag.


\item [[Jean-Yves Girard]]. Geometry of interaction III : accommodating the additives. In Girard, Lafont, and Regnier, editors, Advances in Linear Logic, pages 329 -- 389, Cambridge, 1995. Cambridge University Press.


\item [[Jean-Yves Girard]]. Geometry of interaction IV : the feedback equation. In Stoltenberg-Hansen and V\"a{}\"a{}n\"a{}nen, editors, Logic Colloquium `03, pages 76 -- 117. Association for Symbolic Logic, 2006.


\item [[Jean-Yves Girard]]. Geometry of interaction V : logic in the hyperfinite factor, in memoriam Claire Delaleu (1991-2009). Fully revised version (October 2009). (\href{http://iml.univ-mrs.fr/~girard/GdI5.1.pdf}{pdf})


\item [[Jean-Yves Girard]]. Geometry of Interaction VI: a blueprint for Transcendental Syntax, January 2013, revised August 28th 2013. (\href{http://iml.univ-mrs.fr/~girard/blueprint.pdf}{pdf})



\end{itemize}
Reviews include

\begin{itemize}%
\item [[Jean-Yves Girard]], Part VI of \emph{[[Lectures on Logic]]}


\item Masaru Shirahata, \emph{Geometry of Interaction explained}, (\href{http://www.kurims.kyoto-u.ac.jp/~hassei/algi-13/kokyuroku/19_shirahata.pdf}{pdf})



\end{itemize}
\begin{itemize}%
\item [[Samson Abramsky]], E. Haghverdi and P. Scott, ``Geometry of Interaction and linear combinatory algebra,'' Mathematical Structures in Computer Science (2002), vol. 12, pp. 625-665. (\href{http://www.cs.ox.ac.uk/files/328/ahs.ps}{ps})


\item [[Samson Abramsky]] and R. Jagadeesan, ``New Foundations for the Geometry of Interaction,'' Proceedings 7th Annual IEEE Symp. on Logic in Computer Science, LICS'92, Santa Cruz, CA, USA, 22--25 June 1992. (\href{http://www.cs.ox.ac.uk/files/300/nfgoi.pdf}{pdf})


\item Harry G. Mairson, ``From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple'', Proceedings of FST TCS 2002. (\href{http://www.cs.brandeis.edu/~mairson/Papers/fsttcs02.pdf}{pdf})


\item Linear Logic Wiki, \emph{\href{http://llwiki.ens-lyon.fr/mediawiki/index.php/Geometry_of_interaction}{Geometry of Interaction}}


\item [[Daniel Murfet]], section 5 of \emph{Logic and linear algebra: an introduction} (\href{http://arxiv.org/abs/1407.2650}{arXiv:1407.2650})



\end{itemize}
The operational categorical semantics in [[traced monoidal categories]] is due to

\begin{itemize}%
\item [[André Joyal]], [[Ross Street]], [[Dominic Verity]], \emph{Traced monoidal categories}, Math. Proc. Camb. Phil. Soc. (1996), 119, 447 (\href{http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Esfandir Haghverdi]], \emph{A Categorical Approach to Linear Logic}, Geometry of Proofs and Full Completeness, PhD Thesis, University of Ottawa, Canada 2000.

\end{itemize}
\begin{itemize}%
\item [[Esfandir Haghverdi]], \emph{Unique Decomposition Categories, Geometry of Interaction, and Combinatory Logic, Math. Struc. in Comp. Sci., 10, pp. 205-231.}

\end{itemize}
\begin{itemize}%
\item [[Samson Abramsky]], [[Esfandir Haghverdi]], [[Philip Scott]], \emph{Geometry of Interaction and Linear Combinatory Algebras}. MSCS, vol. 12(5), 2002, 625-665, CUP (2002) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.7818}{citeseer})


\item [[Esfandir Haghverdi]], [[Philip Scott]], \emph{A categorical model for the geometry of interactions} ,Theoretical Computer Science, 2005 (\href{http://xavier.informatics.indiana.edu/~ehaghver/HS-TCS-04.pdf}{pdf})



\end{itemize}
Discussion of [[quantum operations]]/[[completely positive maps]] mentioned above is in

\begin{itemize}%
\item [[Peter Selinger]], \emph{Dagger-compact closed categories and completely positive maps}, Electronic Notes in Theoretical Computer Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 ([[SelingerPositiveMaps.pdf:file]], \href{http://www.mscs.dal.ca/~selinger/papers/dagger.ps}{ps})

\end{itemize}
\begin{itemize}%
\item [[Samson Abramsky]], [[Bob Coecke]], \emph{Physical Traces: Quantum vs. Classical Information Processing}, Electronic Notes in Theoretical Computer Science 69 (2003) (\href{http://arxiv.org/abs/cs/0207057}{arXiv:0207057})

\end{itemize}
\begin{itemize}%
\item [[Bob Coecke]], [[Chris Heunen]], \emph{Pictures of complete positivity in arbitrary dimension}, EPTCS 95, 2012, pp. 27-35 (\href{http://arxiv.org/abs/1110.3055}{arXiv:1110.3055})

\end{itemize}
[[!redirects Geometry of interaction]] [[!redirects geometry of interaction]]

[[!redirects Geometry of Interactions]] [[!redirects geometry of interactions]]



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