String diagram

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

Definition

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 $f:A\to B$ 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 $\alpha :f\Rightarrow g:A\to B$ 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 (on the left hand side) and string diagrams (on the right hand side)

Example

Consider an adjunction $(F,G,\eta ,\varepsilon )$ between two categories ${\mathcal {C}}$ and ${\mathcal {D}}$ where $F:{\mathcal {C}}\leftarrow {\mathcal {D}}$ is left adjoint of $G:{\mathcal {C}}\rightarrow {\mathcal {D}}$ and the natural transformations $\eta :I\rightarrow GF$ and $\varepsilon :FG\rightarrow I$ 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 counit

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:

{\begin{aligned}(\varepsilon F)\circ F(\eta )&=1_{F}\\G(\varepsilon )\circ (\eta G)&=1_{G}\end{aligned}}

The first one is depicted as

Diagrammatic representation of the equality

(\varepsilon F)\circ F(\eta )=1_{F}

Other diagrammatic languages

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

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

References

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.
Selinger, P. (2010). "A Survey of Graphical Languages for Monoidal Categories" (PDF). In Bob Coecke (ed.) (eds.). 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.CS1 maint: uses editors parameter (link)
Joyal, A.; Street, R. (1993). "Braided Tensor Categories". Advances in Mathematics. 102 (1): 20–78. doi:10.1006/aima.1993.1055. ISSN 0001-8708.
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.

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[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.) (eds.). 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.CS1 maint: uses editors parameter (link)

[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.