nLab string diagram

String diagrams

Context

Monoidal categories

monoidal categories

category theory

Applications

Higher category theory

higher category theory

String diagrams

Overview

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

• putting strings next to each other denotes the monoidal product, and having no string at all denotes the unit;

• braiding strings over each other corresponds to – yes, the braiding (if any);

• bending strings around corresponds to dualities on dualizable objects (if any).

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.

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:

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.

Variations

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

• in monoidal categories which are 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 twist operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 360 degrees.

• 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.

• in monoidal categories which are spherical all strings behave as if drawn on a sphere.

• in a hypergraph category, the string diagrams are labeled hypergraphs.

• string diagrams can be extended to represent monoidal functors in several ways. One nice way is described in these slides, and can also be done with “3D regions” as drawn here.

• 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 Poincaré dual to the globular notation: where one uses $d$-dimensional symbols, the other uses $(2-d)$-dimensional symbols.

• string diagrams for bicategories can be generalized to string diagrams for double categories and proarrow equipments by distinguishing between “vertical” and “horizontal” strings.

• Similarly, one can categorify this to “surface diagrams” for 3-categories (including monoidal bicategories) and so on; see for instance here.

• As explained 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 canopolises?.

• 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 (Brady-Trimble 98) and developed in (Ponto-Shulman 12), and was discussed here.

• String diagrams for closed monoidal categories (see also at 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 here and here, but can also be done in essentially the same way as the proof nets used in intuitionistic linear logic; see Lamarche.

• Proof nets for classical linear logic similarly give string diagrams for *-autonomous categories, or more generally linearly distributive categories; see Blute-Cockett-Seely-Trimble.

See the article by Selinger below for more examples.

Introductory material

• John Baez, QG Seminar Fall 2000 (web), Winter 2001 (web), Fall 2006 (web).

• John Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, arXiv

• The Catsters (Simon Willerton), String diagrams (YouTube)

The higher dimensional string diagrams (“zoom complexes” (Kock-Joyal-Batanin-Mascari 07)) used for presenting opetopes in the context of opetopic type theory are introduced in

Software

• 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.

Original articles

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 for compact closed categories

• Max Kelly and M. L. Laplaza, Coherence for compact closed categories. Journal of Pure and Applied Algebra, 19:193–213, 1980.

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:

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.

• Roger Penrose, Applications of negative dimensional tensors , in Combinatorial Mathematics and its Applications, Academic Press (1971) (pdf)

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:

• Peter Freyd, David Yetter, Braided compact closed categories with applications to low dimensional topology Advances in Mathematics, 77:156–182, 1989.

• Peter Freyd and David Yetter, Coherence theorems via knot theory. Journal of Pure and Applied Algebra, 78:49–76, 1992.

• David Yetter, Framed tangles and a theorem of Deligne on braided deformations of tannakian categories In M. Gerstenhaber and Jim Stasheff (eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics 134, pages 325–349. Americal Mathematical Society,

1992.

For more on the history of the notion see the bibliography in (Selinger 09).

Details

String diagrams for monoidal categories are discussed in

For 1-categories in

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

• Andre Joyal, Ross Street and Verity, Traced monoidal categories.

• David I. Spivak, Patrick Schultz, Dylan Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms, arxiv

• John Baez, Quantum Gravity Seminar - Fall 2006. <http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation>
• John Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, arxiv

• Francois Lamarche, Proof Nets for Intuitionistic Linear Logic: Essential nets, 2008 pdf

• Ralf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, pdf

The generalization of string diagrams to one dimension higher is discussed in

The generalization to arbitrary dimension in terms of opetopic “zoom complexes” is due to

Discussion for double categories and pro-arrow equipments is in

• David Jaz Myers, String Diagrams For Double Categories and (Virtual) Equipments (arXiv:1612.02762)