nLab
string diagram

Context

Monoidal categories

monoidal categories

With symmetry

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

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

String diagrams

Idea

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

From Hotz 65

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 quantum computation

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

(…)

References

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

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”.

(Pawel Sobocinski, 2 May 2017)

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.”

(Alex Kissinger, 2 May 2017)

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

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:

See also

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.

See also these:

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

For traced monoidal categories in

For closed monoidal categories in

For biclosed monoidal categories in

For linearly distributive categories in

For indexed monoidal categories in

For symmetric traced monoidal categories in

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

See also at opetopic type theory.

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