# nLab string diagram

String diagrams

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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:

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 Laplaza’s 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

• 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

The generalization to arbitrary dimension in terms of manifold diagrams (generalizing, in particular, opetopic shapes) is due to

Discussion of string diagram calculus for (virtual) double categories and (virtual) pro-arrow equipments:

Discussion of sheet diagrams for rig categories is in

### Software

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

• Eric Finster, Opetopic Diagrams 1 - Basics (video)

• Eric Finster, Opetopic Diagrams 2 - Geometry (video)

• 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.

### Quantum information theory via String diagrams

The observation that a natural language for quantum information theory and quantum computation, specifically for quantum circuit diagrams, is that of string diagrams in †-compact categories (see quantum information theory via dagger-compact categories):

On the relation to quantum logic/linear logic:

Early exposition with introduction to monoidal category theory:

• Bob Coecke, Kindergarten quantum mechanics $[$arXiv:quant-ph/0510032$]$

• Bob Coecke, Introducing categories to the practicing physicist $[$arXiv:0808.1032$]$

• John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone in: New Structures for Physics, Bob Coecke (ed.), Lecture Notes in Physics 813, Springer (2011) 95-174 $[$arxiv/0903.0340$]$

• Bob Coecke, Eric Oliver Paquette, Categories for the practising physicist, in: New Structures for Physics, Lecture Notes in Physics 813, Springer (2010) $[$arXiv:0905.3010, doi:10.1007/978-3-642-12821-9_3$]$

• Bob Coecke, Quantum Picturalism $[$arXiv:0908.1787$]$

Review in contrast to quantum logic:

and with emphasis on quantum computation:

• Jamie Vicary, The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$arXiv:1209.3917, doi:10.1109/LICS.2013.14$]$

Generalization to quantum operations on mixed states (completely positive maps of density matrices):

Formalization of quantum measurement via Frobenius algebra-structures:

Textbook accounts (with background on relevant monoidal category theory):

Refinement to the ZX-calculus for specific control of quantum circuit-diagrams:

Relating the ZX-calculus to braided fusion categories for anyon braiding: