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\begin{document}

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\section*{string diagram}

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

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

[[!include monoidal categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{string_diagrams}{}\section*{{String diagrams}}\label{string_diagrams}

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{relation_to_commutative_diagrams}{Relation to commutative diagrams}\dotfill \pageref*{relation_to_commutative_diagrams} \linebreak
\noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{introductory_material}{Introductory material}\dotfill \pageref*{introductory_material} \linebreak
\noindent\hyperlink{software}{Software}\dotfill \pageref*{software} \linebreak
\noindent\hyperlink{OriginalArticles}{Original articles}\dotfill \pageref*{OriginalArticles} \linebreak
\noindent\hyperlink{surveys}{Surveys}\dotfill \pageref*{surveys} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

\emph{String diagrams} are a graphical calculus for expressing operations in a [[monoidal category]]. The idea is roughly to think of objects in a monoidal category as ``strings'' and a morphism from one tensor product to another as a node which the source strings enter and the target strings exit. Further structure on the monoidal category is encoded in geometrical properties on these strings. For instance

\begin{itemize}%
\item putting strings next to each other denotes the monoidal product, and having no string at all denotes the unit;


\item braiding strings over each other corresponds to -- yes, the [[braided monoidal category|braiding]] (if any);


\item bending strings around corresponds to dualities on [[dualizable object]]s (if any).



\end{itemize}
Many operations in monoidal categories that look rather unenlightening in symbols become very obvious in string diagram calculus, such as the [[trace]]: an output wire gets bent around and connects to an input.

\hypertarget{relation_to_commutative_diagrams}{}\subsection*{{Relation to commutative diagrams}}\label{relation_to_commutative_diagrams}

String diagrams can be seen as dual (in the sense of [[Poincaré duality]]) to commutative diagrams. For instance, in a 2-category, the string diagram for a 2-cell can be obtained as follows:

$\backslash$begin\{centre\} $\backslash$begin\{tikzpicture\}every node/.style=\{node distance=1.5cm\} $\backslash$usetikzlibrary\{calc,automata\}

$\backslash$node (a) \{$A$\}; $\backslash$node\href{b}{above right of=a} \{$B$\}; $\backslash$node\href{c}{right of=b} \{$C$\}; $\backslash$node at ($(b) !.5! (c)$) (bc) \{\}; $\backslash$node\href{d}{below right of=c} \{$D$\}; $\backslash$node\href{e}{below of=bc,node distance=1.875cm} \{$E$\};

$\backslash$draw\href{a}{-{\tt \symbol{62}}} edge nodeabove left \{$f$\} (b); $\backslash$draw\href{b}{-{\tt \symbol{62}}} edge nodeabove \{$g$\} (c); $\backslash$draw\href{c}{-{\tt \symbol{62}}} edge nodeabove right \{$h$\} (d); $\backslash$draw\href{a}{-{\tt \symbol{62}}} edge nodebelow \{$k$\} (e); $\backslash$draw\href{e}{-{\tt \symbol{62}}} edge nodebelow \{$l$\} (d);

$\backslash$node\href{double}{below of=bc,node distance=0.9cm} \{$\backslash$large $\Downarrow \alpha$\};

$\backslash$node\href{egal}{right of=d} \{$\backslash$large $\mapsto$\};

$\backslash$tikzstyle\{point\}=circle,draw,fill,inner sep=1pt; $\backslash$node\href{centre}{right of=egal,node distance=3cm,point} \{\}; $\backslash$draw ($(centre)-(1.5,1.5)$) rectangle ($(centre)+(1.5,1.5)$);

\% Points $\backslash$tikzstyle\{label\}=node distance=0.3cm $\backslash$nodeinner sep=0pt at ($(centre)+(-1,1.52)$) (h) \{\}; $\backslash$nodeinner sep=0pt at ($(centre)+(0,1.52)$) (g) \{\}; $\backslash$nodeinner sep=0pt at ($(centre)+(1,1.52)$) (f) \{\}; $\backslash$nodeinner sep=0pt at ($(centre)+(-0.5,-1.52)$) (l) \{\}; $\backslash$nodeinner sep=0pt at ($(centre)+(0.5,-1.52)$) (k) \{\}; $\backslash$nodeabove of=h,label \{$h$\}; $\backslash$nodeabove of=g,label \{$g$\}; $\backslash$nodeabove of=f,label \{$f$\}; $\backslash$nodebelow of=l,label \{$l$\}; $\backslash$nodebelow of=k,label \{$k$\}; $\backslash$noderight of=centre,label \{$\alpha$\};

\% Edges $\backslash$draw\href{f}{thick} edge\href{centre}{bend left}; $\backslash$draw\href{h}{thick} edge\href{centre}{bend right}; $\backslash$draw\href{l}{thick} edge\href{centre}{bend left}; $\backslash$draw\href{k}{thick} edge\href{centre}{bend right}; $\backslash$draw\href{g}{thick} -- (centre);

\% Regions $\backslash$tikzstyle\{region\}=node distance=0.9cm $\backslash$nodebelow of=centre,region \{$E$\}; $\backslash$nodeleft of=centre,region \{$D$\}; $\backslash$noderight of=centre,region \{$A$\}; $\backslash$nodebelow left of=f,region \{$B$\}; $\backslash$nodebelow right of=h,region \{$C$\}; $\backslash$end\{tikzpicture\}

$\backslash$end\{centre\}

String diagrams for monoidal categories can be obtained in the same way, by considering a monoidal category as a 2-category with a single object.

\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

There are many additional structures on monoidal categories, or similar structures, which can usually be represented by encoding further geometric properties. For instance:

\begin{itemize}%
\item in monoidal categories which are \emph{[[ribbon category|ribbon categories]]} the strings from above behave as if they have a small transversal extension which makes them behave as ribbons. Accordingly, there is a \emph{twist} operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 360 degrees.


\item in a [[traced monoidal category]], the trace can be represented by bending an output string around to connect to an input, even though if the objects are not dualizable the individual ``bends'' do not represent anything.


\item in monoidal categories which are \emph{[[spherical category|spherical]]} all strings behave as if drawn on a sphere.


\item in a [[hypergraph category]], the string diagrams are labeled [[hypergraphs]].


\item string diagrams can be extended to represent [[monoidal functors]] in several ways. One nice way is described in \href{http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf}{these slides}, and can also be done with ``3D regions'' as drawn \href{http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html#c034428}{here}.


\item there is also a string diagram calculus for [[bicategories]], which extends that for monoidal categories regarded as one-object bicategories. Thus, the strings now represent 1-cells and the nodes 2-cells, leaving the two-dimensional planar regions cut out by the strings to represent the 0-cells. This makes it manifest that in general, string diagram notation is \emph{Poincar\'e{} dual} to the [[globe|globular]] notation: where one uses $d$-dimensional symbols, the other uses $(2-d)$-dimensional symbols.


\item string diagrams for bicategories can be generalized to string diagrams for [[double categories]] and [[proarrow equipments]] by distinguishing between ``vertical'' and ``horizontal'' strings.


\item Similarly, one can [[categorification|categorify]] this to ``[[surface diagrams]]'' for [[3-categories]] (including [[monoidal bicategories]]) and so on; see for instance [[toddtrimble:Surface diagrams|here]].


\item As explained \href{http://sbseminar.wordpress.com/2007/07/12/the-operadic-periodic-table/}{here}, in the presence of certain levels of duality it may be better to work with diagrams on cylinders or spheres rather than in boxes. This relates to [[planar algebras]] and [[canopolis|canopolises]].


\item A string diagram calculus for [[monoidal fibrations]] can be obtained as a generalization of C.S. Peirce's ``existential graphs.'' The ideas are essentially contained in (\hyperlink{BradyTrimble98}{Brady-Trimble 98}) and developed in (\hyperlink{PontoShulman12}{Ponto-Shulman 12}), and was discussed \href{http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html}{here}.


\item String diagrams for [[closed monoidal categories]] (see also at \emph{[[Kelly-Mac Lane graph]]}) are similar to those for [[autonomous categories]], but a bit subtler, involving ``boxes'' to separate parts of the diagram. They were used informally by Baez and Stay \hyperlink{BaezQG06}{here} and \hyperlink{Rosetta}{here}, but can also be done in essentially the same way as the [[proof nets]] used in [[intuitionistic logic|intuitionistic]] [[linear logic]]; see \hyperlink{Lamarche08}{Lamarche}.


\item Proof nets for [[classical mathematics|classical]] [[linear logic]] similarly give string diagrams for [[\emph{-autonomous categories]], or more generally [[linearly distributive categories]]; see \hyperlink{BCST}{Blute-Cockett-Seely-Trimble}.}



\end{itemize}
See the article by Selinger below for more examples.

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

\begin{itemize}%
\item [[sharing graph]]


\item [[surface diagram]]


\item [[Kelly-Mac Lane diagram]]


\item [[proof net]]


\item [[Feynman diagram]]


\item [[tensor network]],


\item [[neural network]]


\item [[linguistics|natural language syntax]]



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

\hypertarget{introductory_material}{}\subsubsection*{{Introductory material}}\label{introductory_material}

\begin{itemize}%
\item [[John Baez]], QG Seminar Fall 2000 (\href{http://math.ucr.edu/home/baez/qg-fall2000/}{web}), Winter 2001 (\href{http://math.ucr.edu/home/baez/qg-winter2001/}{web}), Fall 2006 (\href{http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation}{web}).


\item John Baez and Mike Stay, \emph{Physics, Topology, Logic and Computation: A Rosetta Stone}, \href{http://arxiv.org/abs/0903.0340}{arXiv}


\item The Catsters (Simon Willerton), \emph{String diagrams} (\href{http://www.youtube.com/view_play_list?p=50ABC4792BD0A086}{YouTube})



\end{itemize}
The higher dimensional string diagrams (``zoom complexes'' (\hyperlink{KockJoyalBataninMascari07}{Kock-Joyal-Batanin-Mascari 07})) used for presenting [[opetopes]] in the context of [[opetopic type theory]] are introduced in

\begin{itemize}%
\item [[Eric Finster]], \emph{Opetopic Diagrams 1 - Basics} (\href{http://www.youtube.com/watch?v=OANwLohwJqk}{video})


\item [[Eric Finster]], \emph{Opetopic Diagrams 2 - Geometry} (\href{http://www.youtube.com/watch?v=E7OvuA1jRKM}{video})



\end{itemize}
\hypertarget{software}{}\subsubsection*{{Software}}\label{software}

\begin{itemize}%
\item [[Globular]] is a web-based proof assistant for finitely-presented semistrict globular higher categories. It allows one to formalize higher-categorical proofs in finitely-presented n-categories and visualize them as string diagrams.

\end{itemize}
\hypertarget{OriginalArticles}{}\subsubsection*{{Original articles}}\label{OriginalArticles}

[[Günter Hotz]] introduced ``plane nets'' (=string diagrams) and their categories in his 1965 habilitation thesis. This seems to be the first formal definition of string diagrams in the literature.

String diagrams appeared first in [[Max Kelly]] and Laplaza's paper on [[coherence theorem for monoidal categories|coherence for compact closed categories]]

\begin{itemize}%
\item [[Max Kelly]] and M. L. Laplaza, \emph{Coherence for compact closed categories}. Journal of Pure and Applied Algebra, 19:193--213, 1980.

\end{itemize}
Where in Kelly-Laplaza do string diagrams appear? I can't find any picture of a string diagram in the paper. Perhaps they are described somewhere in the text, but I can't see it.

and again in [[Ross Street]]`s work with [[André Joyal]] in the mid-80's:

\begin{itemize}%
\item [[Andre Joyal]] and [[Ross Street]], \emph{The geometry of tensor calculus I}, Advances in Math. 88 (1991) 55-112; MR92d:18011. (\href{http://tqft.net/other-papers/Geometry%20of%20Tensor%20Calculus%20-%20Joyal%20&%20Street.pdf}{pdf})


\item [[Andre Joyal]] and [[Ross Street]], \emph{The geometry of tensor calculus II}. (\href{http://www.math.mq.edu.au/~street/GTCII.pdf}{pdf}.



\end{itemize}
An early amplification of the use of string-diagram notation as an alternative for the traditional index-calculus for [[tensors]] is due to [[Roger Penrose]].

\begin{itemize}%
\item [[Roger Penrose]], \emph{Applications of negative dimensional tensors} , in Combinatorial Mathematics and its Applications, Academic Press (1971) (\href{http://www2.math.uic.edu/~kauffman/Penrose.pdf}{pdf})

\end{itemize}
Probably [[David Yetter]] was the first (at least in public) to write them with ``coupons'' (a term used by [[Nicolai Reshitikhin]] and [[Turaev]] a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure. See also these:

\begin{itemize}%
\item [[Peter Freyd]], [[David Yetter]], \emph{Braided compact closed categories with applications to low dimensional topology} Advances in Mathematics, 77:156--182, 1989.


\item [[Peter Freyd]] and [[David Yetter]], \emph{Coherence theorems via knot theory}. Journal of Pure and Applied Algebra, 78:49--76, 1992.


\item [[David Yetter]], \emph{Framed tangles and a theorem of Deligne on braided deformations of tannakian categories} In M. Gerstenhaber and [[Jim Stasheff]] (eds.) \emph{Deformation Theory and Quantum Groups with Applications to Mathematical Physics}, Contemporary Mathematics 134, pages 325--349. Americal Mathematical Society, 1992.



\end{itemize}
For more on the history of the notion see the bibliography in (\hyperlink{Selinger}{Selinger 09}).

\hypertarget{surveys}{}\subsubsection*{{Surveys}}\label{surveys}

\begin{itemize}%
\item [[Peter Selinger]], \emph{A survey of graphical languages for monoidal categories} (\href{http://arxiv.org/abs/0908.3347}{arXiv:0908.334})

\end{itemize}
\begin{itemize}%
\item [[Ross Street]], \href{http://www.mta.ca/~cat-dist/CT95Docs/LowDim.ps}{Low dimensional topology and higher-order categories} -- talks about surface diagrams and includes some pictures (PS version only).


\item Ross Street, ``Categorical structures'' -- discusses string diagrams for bicategories.


\item Some philosophical discussion is given in section 10.4 of [[David Corfield]], \emph{Towards a Philosophy of Real Mathematics}, CUP, 2003.



\end{itemize}
\hypertarget{details}{}\subsubsection*{{Details}}\label{details}

String diagrams for [[monoidal categories]] are discussed in

\begin{itemize}%
\item [[Andre Joyal]] and [[Ross Street]], \emph{The geometry of tensor calculus I}, Advances in Math. 88 (1991) 55-112; MR92d:18011. (\href{http://tqft.net/other-papers/Geometry%20of%20Tensor%20Calculus%20-%20Joyal%20&%20Street.pdf}{pdf})


\item [[Andre Joyal]] and [[Ross Street]], \emph{The geometry of tensor calculus II}. (\href{http://www.math.mq.edu.au/~street/GTCII.pdf}{pdf}.


\item [[Andre Joyal]] and [[Ross Street]], \emph{Planar diagrams and tensor algebra}, available \href{http://www.math.mq.edu.au/~street/PlanarDiags.pdf}{here}.



\end{itemize}
For [[1-categories]] in

\begin{itemize}%
\item [[Dan Marsden]], \emph{Category Theory Using String Diagrams}, (\href{https://arxiv.org/abs/1401.7220}{arXiv:1401.7220}).

\end{itemize}
(therein: many explicit calculations, colored illustrations, avoiding the common practice of indicating 0-cells by non-filled circles)

For [[traced monoidal categories]] in

\begin{itemize}%
\item [[Andre Joyal]], [[Ross Street]] and [[Dominic Verity|Verity]], \emph{Traced monoidal categories}.


\item David I. Spivak, Patrick Schultz, Dylan Rupel, \emph{String diagrams for traced and compact categories are oriented 1-cobordisms}, \href{https://arxiv.org/abs/1508.01069}{arxiv}



\end{itemize}
For [[closed monoidal categories]] in

\begin{itemize}%
\item [[John Baez]], Quantum Gravity Seminar - Fall 2006. $<$http://math.ucr.edu/home/baez/qg-fall2006/index.html\#computation{\tt \symbol{62}}

\end{itemize}
\begin{itemize}%
\item [[John Baez]] and [[Mike Stay]], \emph{Physics, Topology, Logic and Computation: A Rosetta Stone}, \href{https://arxiv.org/abs/0903.0340}{arxiv}


\item Francois Lamarche, \emph{Proof Nets for Intuitionistic Linear Logic: Essential nets}, 2008 \href{http://hal.inria.fr/docs/00/34/73/36/PDF/prfnet1.pdf}{pdf}



\end{itemize}
\begin{itemize}%
\item Ralf Hinze, \emph{Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick}, \href{http://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf}{pdf}

\end{itemize}
For [[biclosed monoidal categories]] in

\begin{itemize}%
\item [[Bob Coecke]], Edward Grefenstette, and [[Mehrnoosh Sadrzadeh]], \emph{Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus}, 2013 \href{https://www.sciencedirect.com/science/article/pii/S0168007213000626}{link}

\end{itemize}
For [[linearly distributive categories]] in

\begin{itemize}%
\item [[Richard Blute]] and [[Robin Cockett]] and [[R.A.G. Seely]] and [[Todd Trimble]], \emph{Natural deduction and coherence for weakly distributive categories.}

\end{itemize}
For [[indexed monoidal categories]] in

\begin{itemize}%
\item Geraldine Brady, [[Todd Trimble]], \emph{[[A string diagram calculus for predicate logic]]} (1998)


\item [[Kate Ponto]], [[Michael Shulman]], \emph{Duality and traces for indexed monoidal categories}, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (\href{http://arxiv.org/abs/1211.1555}{arXiv:1211.1555})



\end{itemize}
The generalization of string diagrams to one dimension higher is discussed in

\begin{itemize}%
\item [[Todd Trimble]], \emph{\href{/toddtrimble/published/Surface+diagrams}{Surface diagrams}}


\item [[John Barrett]], [[Catherine Meusburger]], [[Gregor Schaumann]], \emph{Gray categories with duals and their diagrams}, available \href{http://arxiv.org/abs/1211.0529}{here}.



\end{itemize}
The generalization to arbitrary dimension in terms of [[opetope|opetopic]] ``zoom complexes'' is due to

\begin{itemize}%
\item [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], \emph{Polynomial functors and opetopes} (\href{http://arxiv.org/abs/0706.1033}{arXiv:0706.1033})

\end{itemize}
Discussion for [[double categories]] and [[2-category equipped with proarrows|pro-arrow equipments]] is in

\begin{itemize}%
\item David Jaz Myers, \emph{String Diagrams For Double Categories and (Virtual) Equipments} (\href{https://arxiv.org/abs/1612.02762}{arXiv:1612.02762})

\end{itemize}
See also at \emph{[[opetopic type theory]]}.

[[!redirects string diagrams]] [[!redirects circuit diagram]] [[!redirects circuit diagrams]] [[!redirects graphical calculus]]



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