String diagram


In category theory, string diagrams are a way of representing morphisms in monoidal categories, or more generally 2-cells in 2-categories.

Definition

The idea is to represent structures of dimension d by structures of dimension 2-d, using Poincaré duality. Thus,
The parallel composition of 2-cells corresponds to the horizontal juxtaposition of diagrams and the sequential composition to the vertical juxtaposition of diagrams.

Example

Consider an adjunction between two categories and where is left adjoint of and the natural transformations and are respectively the unit and the counit. The string diagrams corresponding to these natural transformations are:
The string corresponding to the identity functor is drawn as a dotted line and can be omitted.
The definition of an adjunction requires the following equalities:
The first one is depicted as

Other diagrammatic languages

Morphisms in monoidal categories can also be drawn as string diagrams since a strict monoidal category can be seen as a 2-category with only one object and Mac Lane's strictification theorem states that any monoidal category is monoidally equivalent to a strict one. The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as braided monoidal categories, dagger categories, etc. and is related to geometric presentations for braided monoidal categories and ribbon categories. In quantum computing, there are several diagrammatic languages based on string diagrams for reasoning about linear maps between qubits, the most well-known of which is the ZX-calculus.