to node\[zxnode=, pos=0.25,scale=1.2\][$a$]{} node\[zxnode=, pos=0.75,scale=1.2\] [$b$]{}(0,2);

    

(1.5,0) to node\[zxnode=, pos=0.25,scale=1.2\][$b$]{} node\[zxnode=, pos=0.75,scale=1.2\] [$a$]{}(0,2);

The two free-floating vertices on the left can move around each other, whereas on the right they are confined to move on a line and do not commute.[^4] In other words, adding a wire to a diagram can turn a commutative situation into a noncommutative one.

Taking this idea seriously leads to the following sketch of a programme. By Gelfand duality, a combinatorial theory, such as the theory of finite sets and functions or finite graphs and graph homomorphisms, can be expressed in terms of finite-dimensional commutative algebra and therefore represented in the graphical calculus. To quantise the objects of the theory — in our case, sets or graphs — one simply passes from commutative to noncommutative algebras. The novelty of our approach lies in our treatment of morphisms: having formulated the theory in term of string diagrams, we quantise morphisms by adding a wire through the corresponding vertices. We illustrate this idea with an example:

\[eq:quantisationexample\]

(0,0) to (0,2) to \[out=135\] (-0.75,3); (0,2) to \[out=45\] (0.75, 3); at (0,1) [$f$]{}; at (0,2) ;

   =   

(0,0) to (0,0.75) to \[out=135\] (-0.75,1.75) to (-0.75,3); (0,0.75) to \[out=45\] (0.75, 