String diagram
Encyclopedia
In category theory
, string diagrams are a way of representing 2-cells in 2-categories. The idea is to represent structures of dimension d by structures of dimension 2-d, using the Poincaré duality
. Thus,
For example, consider an adjunction
between two categories C and D where 
is left adjoint of 
and the natural transformations 
and 
are respectively the unit and the counit. The string diagram corresponding to the counit is the following:
From left to right and up to down, the three areas correspond respectively to the objects D, C and D, the three segments (excluding the border) to the functors G, F and I (the dotted one), and the intersection in the middle to the natural transformation
.
The horizontal composition corresponds to the horizontal juxtaposition of two diagrams and the vertical composition to the vertical composition of two diagrams.
Monoidal categories
can also be pictured this way since they can be seen as 2-categories with only one object (there will therefore be only one type of plane).
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, string diagrams are a way of representing 2-cells in 2-categories. The idea is to represent structures of dimension d by structures of dimension 2-d, using the Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
. Thus,
- an object is represented by a portion of plane,
- a 1-cell
is represented by a vertical segment — called a string — separating the plane in two (the left part corresponding to A and the right one to B), - a 2-cell
is represented by an intersection of strings (the strings corresponding to f above the link, the strings corresponding to g below the link).
For example, consider an adjunction
between two categories C and D where is left adjoint of and the natural transformations and are respectively the unit and the counit. The string diagram corresponding to the counit is the following:From left to right and up to down, the three areas correspond respectively to the objects D, C and D, the three segments (excluding the border) to the functors G, F and I (the dotted one), and the intersection in the middle to the natural transformation
.- TODO: zig-zag identities
The horizontal composition corresponds to the horizontal juxtaposition of two diagrams and the vertical composition to the vertical composition of two diagrams.
Monoidal categories
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
can also be pictured this way since they can be seen as 2-categories with only one object (there will therefore be only one type of plane).