nLab surface diagram

Idea

Surface diagrams categorify string diagrams: while string diagram calculus describes the composition of morphisms in 2-categories, surface diagram calculus describes composition of morphisms in (non-strict) 3-categories.

Variants

There are several discussions and variation of surface diagrams available.

• Trimble discusses surface diagrams via stratified Morse theory,
• Barret-Meusburger-Schaumann define a notion of surface diagrams for Gray categories with duals,
• Hummon defines a notion for Gray categories,
• Dorn-Douglas define a notion of manifold diagram, which specializes to surface diagrams in dimension 3.

While all of the above notions are “in essence” equivalent, they differ in some of their technical details.

Related concepts

• manifold diagram
• string diagram
• tangle

References

• Todd Trimble, Surface diagrams (2010)

• Benjamin Hummon. Surface diagrams for gray-categories, Phd Thesis, UC San Diego (2012) [escholar:5b24s9cc, pdf]

• John Barrett, Catherine Meusburger, Gregor Schaumann: Gray categories with duals and their diagrams, Advances in Mathematics 450 (2024) 109740 [arXiv:1211.0529, doi:10.1016/j.aim.2024.109740]

• Christoph Dorn, Christopher Douglas: Manifold diagrams and tame tangles (2022) [arXiv:2208.13758]

  (via manifold diagrams)