String Diagram

A string diagram is the two-dimensional graphical calculus of a monoidal category: an object is a wire, a morphism a box with an -wire in and a -wire out, sequential composition is vertical stacking, the tensor is horizontal juxtaposition, the unit is the empty diagram, and the symmetry is a crossing of wires. Reading a diagram top-to-bottom recovers a formula; a type mismatch is a diagram that will not connect.

Theorem. Diagrams commute iff they are isotopic

The coherence theorem for (symmetric) monoidal categories says two morphism expressions are equal by the axioms iff their diagrams are related by a planar (resp. spatial) isotopy that fixes the boundary. Bureaucratic bookkeeping of associators and unitors becomes “the picture can be deformed”, so a diagram is an equivalence class of formulas.

Example. Copy and discard

In a Markov category each wire carries a comonoid, where is a branching node, a terminating node, and their laws are local moves on the picture. Extra structure earns extra vocabulary. Caps and cups mark duals, functor boxes a monoidal functor’s image, and stripe and tube diagrams monads and bialgebras, each sitting one level up from ordinary wires.

References

• Andre Joyal, Ross Street, “The geometry of tensor calculus I”, Adv. Math. 88 (1991)
• Peter Selinger, “A survey of graphical languages for monoidal categories” (2010)