String diagram

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In category theory, string diagrams are a way of representing morphisms in monoidal categories, or more generally 2-cells in 2-categories.


The idea is to represent structures of dimension d by structures of dimension 2-d, using Poincaré duality. Thus,

  • an object is represented by a portion of plane,
  • a 1-cell is represented by a vertical segment—called a string—separating the plane in two (the right part corresponding to A and the left one to B),
  • a 2-cell is represented by an intersection of strings (the strings corresponding to f above the link, the strings corresponding to g below the link).

The parallel composition of 2-cells corresponds to the horizontal juxtaposition of diagrams and the sequential composition to the vertical juxtaposition of diagrams.

Duality between commutative diagrams and string diagrams.
Duality between commutative diagrams (on the left hand side) and string diagrams (on the right hand side)


Consider an adjunction between two categories and where is left adjoint of and the natural transformations and are respectively the unit and the counit. The string diagrams corresponding to these natural transformations are:

String diagram of the unit
String diagram of the unit
String diagram of the counit
String diagram of the counit
String diagram of the identity 2-cell
String diagram of the identity

The string corresponding to the identity functor is drawn as a dotted line and can be omitted. The definition of an adjunction requires the following equalities:

The first one is depicted as

Diagrammatic representation of the equality
Diagrammatic representation of the equality

Other diagrammatic languages[edit]

Morphisms in monoidal categories can also be drawn as string diagrams [1] since a strict monoidal category can be seen as a 2-category with only one object (there will therefore be only one type of planar region) and Mac Lane's strictification theorem states that any monoidal category is monoidally equivalent to a strict one. The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as braided monoidal categories, dagger categories,[2] etc. and is related to geometric presentations for braided monoidal categories[3] and ribbon categories.[4] In quantum computing, there are several diagrammatic languages based on string diagrams for reasoning about linear maps between qubits, the most well-known of which is the ZX-calculus.

External links[edit]

  • TheCatsters (2007). String diagrams 1 (streamed video). Youtube.
  • String diagrams in nLab


  1. ^ Joyal, André; Street, Ross (1991). "The geometry of tensor calculus, I" (PDF). Advances in Mathematics. 88 (1): 55–112. doi:10.1016/0001-8708(91)90003-P. ISSN 0001-8708.
  2. ^ Selinger, P. (2010). "A Survey of Graphical Languages for Monoidal Categories" (PDF). In Bob Coecke (ed.). New Structures for Physics. Lecture Notes in Physics. 813. Springer Berlin Heidelberg. pp. 289–355. arXiv:0908.3347. Bibcode:2009arXiv0908.3347S. doi:10.1007/978-3-642-12821-9_4. ISBN 978-3-642-12820-2.
  3. ^ Joyal, A.; Street, R. (1993). "Braided Tensor Categories". Advances in Mathematics. 102 (1): 20–78. doi:10.1006/aima.1993.1055. ISSN 0001-8708.
  4. ^ Shum, Mei Chee (1994-04-11). "Tortile tensor categories". Journal of Pure and Applied Algebra. 93 (1): 57–110. doi:10.1016/0022-4049(92)00039-T. ISSN 0022-4049.

External links[edit]