About the history of "**string diagrams**" in monoidal categories:

The now called "string diagrams" as a topological representation of morphisms in (free) monoidal categories were introduced in 1965 under the notion "plane nets" by Günter Hotz in his habilitation thesis:

Günter Hotz: *Eine Algebraisierung des Syntheseproblems von Schaltkreisen I+II*

Elektronische Informationsverarbeitung und Kybernetik (EIK), Band 1, Heft 3+4, 1965, Akademie-Verlag Berlin

Download links:

https://www.magentacloud.de/lnk/LiPMlYfh (part 1)

https://www.magentacloud.de/lnk/YivslUWJ (part 2)

*Abstract:*

The composition of switching circuits A, B, to new circuits can be reduced to two basic operations:

The sequential composition A \circ B of A and B, if the number of outputs of is equal to the number of inputs of A.

The composition of A and B to A \times B is analogue to the cartesian product. The number of inputs of A \times B is equal to the sum of the numbers of the inputs of A and B. The same holds for the outputs.

Between different compositions an equivalence is introduced which is compatible with \circ and \times. When the compositions are carried over to the set of the equivalence classes, one gets an algebraic structure that forms a category respective to \circ and a semi-group (monoid) respective \times. The compositions satisfy the relation

(A \times B) \circ (C \times D) = (A \circ C) \times (B \circ D)

if A \circ C and B \circ D are defined.

A category with this property is called an x-category.

The categories F studied here can be characterized in the following way:

- The set of units of F form a semi-group generated by one generator relative to \times.
- F has a countable generator set or it holds
- F is a subcategory of a category satisfying 1. and 2.

In a free x-category of this type holds the cancellation law relative to \circ and \times under a weak condition.

In the whole theory of the x-category, ''plane nets'', a concept from the combinatorial topology related to the braids of Artin plays an important role.

Each of the studied free x-categories may be mapped onto an x-category of plane nets by a functor. Each theorem about F by the functor is carried over to a theorem about a category of nets. The theorem proved for the nets can easily be proved in most cases for F; i.e. the theorems about F have an essential combinatorial character. This is the reason that the first chapter only deals about nets.

The connection of the switching circuits represented by the elements of F with their function gives a functor from F into the x-category C(S) of the maps of type f: S^n \to S^m, where S is a countable set containing at least two elements. The \times product is in C(S) the cartesian product.*

-- end abstract

Prof. Hotz is one of the "fathers" of computer science in Germany. He has a strong mathematical background in topology and wrote his dissertation on a topic from Knot theory under supervision of Kurt Reidemeister in Göttingen.

8more comments