# nLab string diagram

String diagrams

### Context

#### Monoidal categories

monoidal categories

category theory

## Applications

#### Higher category theory

higher category theory

# String diagrams

## Idea

String diagrams constitute a graphical calculus for expressing operations in monoidal categories.

In the archetypical cases of the Cartesian monoidal category of finite sets this is Hotz’s notation (Hotz 65) for automata, while for finite-dimensional vector spaces with their usual tensor product this is Penrose’s notation (Penrose 71a, Penrose-Rindler 84) for tensor networks; but the same idea immediately applies more generally to any other monoidal category and yet more generally to bicategories, etc.

The idea is roughly to think of objects in a monoidal category as “strings” and of morphisms 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:

Many operations in monoidal categories that look unenlightening in symbols become obvious in string diagram calculus, such as the trace: an output wire gets bent around and connects to an input.

String diagrams may be seen as dual (in the sense of Poincaré duality) to commutative diagrams. For instance, in a 2-category, an example of a string diagram for a 2-morphism (shown on the left) is shown on the right here:

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.

## Variants

There are many additional extra structures on monoidal categories, or similar structures, which can usually be represented by encoding further geometric properties of their string diagram calculus 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 multiplicative linear logic with negation similarly give string diagrams for *-autonomous categories, or more generally linearly distributive categories; see Blute-Cockett-Seely-Trimble.

• Proof surfaces for noncommutative multiplicative linear logic with negation, see Dunn-Vicary

• Sheet diagrams, string diagrams drawn on a branching surface, may be used for rig categories, see Comfort-Delpeuch-Hedges.

See also Selinger 09 for a review of different string diagram formalisms.

## Examples

### In linear algebra

String diagram calculus in linear algebra:

### In representation theory

String diagram calculus in representation theory: Mandula 81, Cvitanović 08

### In Lie theory

For string diagrams calculus in Lie theory see at:

### In perturbative quantum field theory

For applications of string diagram calculus in perturbative quantum field theory, see at

(…)

### Introduction and survey

Discussion in representation theory:

Introductions to and surveys of string diagram calculus:

From the point of view of finite quantum mechanics in terms of dagger-compact categories:

From the point of view of tensor networks in solid state physics:

Some philosophical discussion is given in

• David Corfield, Section 10.4 of: Towards a Philosophy of Real Mathematics, CUP, 2003.

### Original articles

The development and use of string diagram calculus pre-dates its graphical appearance in print, due to the difficulty of printing non-text elements at the time.

Many calculations in earlier works were quite clearly worked out with string diagrams, then painstakingly copied into equations. Sometimes, clearly graphical structures were described in some detail without actually being drawn: e.g. the construction of free compact closed categories in Kelly and Laplazas 1980 “Coherence for compact closed categories”.

This idea that string diagrams are, due to technical issues, only useful for private calculation, is said explicitly by Penrose. Penrose and Rindler’s book “Spinors and Spacetime” (CUP 1984) has an 11-page appendix full of all sorts of beautiful, carefully hand-drawn graphical notation for tensors and various operations on them (e.g. anti-symmetrization and covariant derivative). On the second page, he says the following:

“The notation has been found very useful in practice as it grealy simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way.”

The first formal definition of string diagrams in the literature appears to be in

• Günter Hotz, Eine Algebraisierung des Syntheseproblems von Schaltkreisen, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 (part I, part II, pdf)

Application of string diagrams to tensor-calculus in mathematical physics (hence for the case that the ambient monoidal category is that of finite dimensional vector spaces equipped with the tensor product of vector spaces) was propagated by Roger Penrose, whence physicists know string diagrams as Penrose notation for tensor calculus:

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

• Roger Penrose, Angular momentum: An approach to combinatorial spacetime, in Ted Bastin (ed.) Quantum Theory and Beyond, Cambridge University Press (1971), pp.151-180 (pdf)

• Roger Penrose, On the nature of quantum geometry, in: J. Klauder (ed.) Magic Without Magic, Freeman, San Francisco, 1972, pp. 333–354 (spire:74082, pdf)

• Roger Penrose, Wolfgang Rindler, appendix (p. 424-434) of: Spinors and space-time – Volume 1: Two-spinor calculus and relativistic fields, Cambridge University Press 1984 (doi:10.1017/CBO9780511564048)

From the point of view of monoidal category theory, an early description of string diagram calculus (without actually depicting any string diagrams, see the above comments) is in:

following

and in

String diagram calculus was apparently popularized by its use in

Probably David Yetter was the first (at least in public) to write string diagrams with “coupons” (a term used by Nicolai Reshetikhin and Turaev a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure.

• 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.

• Paul-André Melliès, Functorial boxes in string diagrams, Procceding of Computer Science Logic 2006 in Szeged, Hungary. 2006 (hal:00154243, pdf, pdf)

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

• Dan Marsden, Category Theory Using String Diagrams, (arXiv:1401.7220).

(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
• George Kaye, The Graphical Language of Symmetric Traced Monoidal Categories, (arXiv:2010.06319)

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)