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The | In [[category theory]], '''string diagrams''' are a way of representing 2-cells in [[2-categories]]. The idea is to represent structures of dimension ''d'' by structures of dimension ''2-d'', using the [[Poincaré duality]]. Thus, | ||
* an object is represented by a portion of plane, | |||
* a 1-cell <math>f:A\to B</math> is represented by a vertical segment — called a ''string'' — separating the plane in two (the left part corresponding to ''A'' and the right one to ''B''), | |||
* a 2-cell <math>\alpha:f\Rightarrow g:A\to B</math> is represented by an intersection of strings (the strings corresponding to ''f'' above the link, the strings corresponding to ''g'' below the link). | |||
For example, consider an [[Adjoint functors|adjunction]] <math>(F,G,\eta,\varepsilon)</math> between two categories ''C'' and ''D'' where <math>F:C\to D</math> is left adjoint of <math>G:D\to C</math> and the natural transformations <math>\eta:I\rightarrow GF</math> and <math>\varepsilon:FG\rightarrow I</math> are respectively the unit and the counit. The string diagram corresponding to the counit is the following: | |||
[[Image:String diag unit.png|center]] | |||
From left to right and up to down, the three areas correspond respectively to the objects ''D'', ''C'' and ''D'', the three segments (excluding the border) to the functors ''G'', ''F'' and ''I'' (the dotted one), and the intersection in the middle to the natural transformation <math>\varepsilon</math>. | |||
:''TODO: zig-zag identities'' | |||
The horizontal composition corresponds to the horizontal juxtaposition of two diagrams and the vertical composition to the vertical composition of two diagrams. | |||
[[monoidal category|Monoidal categories]] can also be pictured this way<ref>A. Joyal and R. Street. Geometry of Tensor Calculus I, Advances in Mathematics, 1991.</ref> since they can be seen as 2-categories with only one object (there will therefore be only one type of plane). 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#Symmetric_monoidal_categories|symmetric monoidal categories]], [[dagger categories]],<ref>P. Selinger. A survey of graphical languages for monoidal categories. New Structures for Physics 2007.</ref> and is related to geometric presentations for [[braided monoidal categories]]<ref>Joyal and Street. Braided tensor categories. Advances in Mathematics, 1993.</ref> and [[Braided_monoidal_categories#Ribbon_categories|ribbon categories]].<ref>Mei Chee Shum. Tortile tensor categories. Journal of Pure and Applied Algebra, 1994.</ref> | |||
== External links == | |||
* {{cite video |people=TheCatsters |title=String diagrams 1 |date=2007 |format=streamed video |publisher=Youtube |url=http://www.youtube.com/watch?v=USYRDDZ9yEc&feature=related}} | |||
== References == | |||
{{reflist}} | |||
{{DEFAULTSORT:String Diagram}} | |||
[[Category:Higher category theory]] | |||
[[Category:Monoidal categories]] | |||
{{categorytheory-stub}} | |||
Revision as of 11:51, 22 May 2013
In category theory, string diagrams are a way of representing 2-cells in 2-categories. The idea is to represent structures of dimension d by structures of dimension 2-d, using the 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 left part corresponding to A and the right 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).
For example, consider an adjunction between two categories C and D where is left adjoint of and the natural transformations and are respectively the unit and the counit. The string diagram corresponding to the counit is the following:
From left to right and up to down, the three areas correspond respectively to the objects D, C and D, the three segments (excluding the border) to the functors G, F and I (the dotted one), and the intersection in the middle to the natural transformation .
- TODO: zig-zag identities
The horizontal composition corresponds to the horizontal juxtaposition of two diagrams and the vertical composition to the vertical composition of two diagrams.
Monoidal categories can also be pictured this way[1] since they can be seen as 2-categories with only one object (there will therefore be only one type of plane). The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as symmetric monoidal categories, dagger categories,[2] and is related to geometric presentations for braided monoidal categories[3] and ribbon categories.[4]
External links
References
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- ↑ A. Joyal and R. Street. Geometry of Tensor Calculus I, Advances in Mathematics, 1991.
- ↑ P. Selinger. A survey of graphical languages for monoidal categories. New Structures for Physics 2007.
- ↑ Joyal and Street. Braided tensor categories. Advances in Mathematics, 1993.
- ↑ Mei Chee Shum. Tortile tensor categories. Journal of Pure and Applied Algebra, 1994.