UC San Diego

We exhibit a calculus of dot, string, and surface diagrams for describing computadic compositions. A surface diagram is a cube equipped with a stratification of regions, walls, seams, and nodes. We label the strata with morphisms in a strict 3-category C by codimension. Away from nodes, horizontal slice stratified squares are string diagrams representing compositions of 2-morphisms. In particular, a surface diagram has source and target string diagrams on its bottom and top faces, respectively. We may evaluate a surface diagram to give a 3-morphism in C. The domain and codomain are given by the evaluation of the source and target string diagrams. We build a 3-category Sd(C) of surface diagrams labeled by C in which composition is given by gluing along common faces. More generally, there is a 3-category Sd(G) of surface diagrams for any 3- computad G of generators. We characterize Sd(G) as the free 3-category on G. In Sd(G), diagrams are taken up to an isotopy-like relation called evolution. This relation destroys the braiding that we expect to have in a tricategory. We wish to capture braiding without working in the fully weak setting of a tricategory. We instead choose to work with in the semi-strict setting of Gray- categories. We define Gray surface diagrams to be those with certain good projection properties. Working up to an appropriate form of evolution, we construct the free Gray- category SdGray(G) of Gray-surface diagrams on a Gray- category. Last, we demonstrate the utility of surface diagrams by studying coherence relations for equivalence in a Gray-category. We show that the data encoding any incoherent equivalence may be recast as a coherent equivalence