String diagram

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 $f:A\to B$ 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 $\alpha:f\Rightarrow g:A\to B$ is represented by an intersection of strings (the strings corresponding to f above the link, the strings corresponding to g below the link).

For example, consider an adjunction $(F,G,\eta,\varepsilon)$ between two categories C and D where $F:C\to D$ is left adjoint of $G:D\to C$ and the natural transformations $\eta:I\rightarrow GF$ and $\varepsilon:FG\rightarrow I$ 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 $\varepsilon$.

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 since they can be seen as 2-categories with only one object (there will therefore be only one type of plane).

External links

• TheCatsters (2007) (streamed video). String diagrams 1. Youtube. http://www.youtube.com/watch?v=USYRDDZ9yEc&feature=related.