A syntax for string diagrams

[mroman42]

What syntax should we use to represent string diagrams internally?

My first idea is to write the diagrams in terms of vertical and horizontal compositions of natural transformations. Every natural transformation can be seen as a rectangle of the diagram with a number of "inputs" and "outputs". And those rectangles can be composed vertically and horizontally.

As an example, let's take the string diagram for an adjuntion

This can be seen as the vertical composition of:

• The horizontal composition of these two:
  
• And the horizontal composition of these two:
  

Where the identity transformations have 1 input and 1 output, while epsilon and eta have 2 inputs and 1 output and 1 input and 2 outputs respectively. We are not drawing the identities. In the proposed syntax this would translate to:

(F * ε) o (η * F)

which is equal to F (the identity natural transformation in the functor F) if we read it as an equation.

[eparejatobes]

Yup. The fun starts when you need to work with all that is still implicit in what you write there: ε has arity (2,1), these input/output things are typed, ...

Recommendations:

• Open graphs and monoidal theories which is actually the same technology (cospans of labeled (monoidal) graphs with discrete source and target) as in
• Calculating colimits compositionally

The actual reason all this works is that cospans (or spans) contain a lot of free monoidal blablabla constructions.

[eparejatobes]

This is a good read for getting some context

• Graphical languages - RFC Walters

[mroman42]

I see. I'm going to read the papers.