String diagram

String diagrams are a convenient notation for depicting 0-cells, 1-cells, and 2-cells in a bicategory, and in particular, objects and morphisms in a monoidal category (this is a special case called a Single faced string diagram):

a 0-cell is depicted as a face (in the case of a monoidal category there is only a single 0-cell)
a 1-cell $𝑓 : 𝑋 \to 𝑌$ is depicted by a line with $𝑋$ on the right and $𝑌$ on the left;
a 2-cell $𝛽 : 𝑓 \Rightarrow 𝑔 : 𝑋 \to 𝑌$ is depicted as a node with $𝑓$ coming out the bottom and $𝑔$ coming out the top.

Horizontal composition is represented by horizontal juxtaposition, and vertical composition is represented by vertical juxtaposition.

String diagrams are often described as Poincaré dual to their counterpart in commutative diagrams. In the former 0-cells are points, in the latter they are faces, &c.

Bibliography

2011. Physics, topology, logic and computation: A Rosetta stone
1991. The geometry of tensor calculus, I
1996. Categorical structures

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