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- String_diagram abstract "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, 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 identitiesThe 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). The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as symmetric monoidal categories, dagger categories, and is related to geometric presentations for braided monoidal categories and ribbon categories.".
- String_diagram thumbnail String_diag_unit.png?width=300.
- String_diagram wikiPageExternalLink watch?v=USYRDDZ9yEc&feature=related.
- String_diagram wikiPageID "3967296".
- String_diagram wikiPageRevisionID "558800839".
- String_diagram hasPhotoCollection String_diagram.
- String_diagram subject Category:Higher_category_theory.
- String_diagram subject Category:Monoidal_categories.
- String_diagram type Abstraction100002137.
- String_diagram type Class107997703.
- String_diagram type Collection107951464.
- String_diagram type Group100031264.
- String_diagram type MonoidalCategories.
- String_diagram comment "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.".
- String_diagram label "String diagram".
- String_diagram sameAs m.0b93dt.
- String_diagram sameAs Q7623974.
- String_diagram sameAs Q7623974.
- String_diagram sameAs String_diagram.
- String_diagram wasDerivedFrom String_diagram?oldid=558800839.
- String_diagram depiction String_diag_unit.png.
- String_diagram isPrimaryTopicOf String_diagram.
