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.
|Commitee:||Intriligator, Ken, Lerner, Sorin, Rogalski, Dan, Wenzl, Hans|
|School:||University of California, San Diego|
|School Location:||United States -- California|
|Source:||DAI-B 74/03(E), Dissertation Abstracts International|
|Keywords:||Coherence, Equivalence, Gray-categories, Surface diagrams|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be