Cartesian Gray-Monoidal Double Categories
| Autor: | Morehouse, Edward |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally globular, Gray categories. The motivating example comprises double categories themselves, together with their functors, transformations, and modifications. A one-object locally cubical Gray category is a Gray-monoidal double category. Braiding, syllepsis, and symmetry for these is introduced in a manner analogous to that for 2-categories. Adding cartesian structure requires the introduction of doubly-lax functors of double categories to manage the order of copies. The resulting theory is algebraically rather complex, largely due to the bureaucracy of linearizing higher-dimensional boundary constraints. Fortunately, it has a relatively simple and compelling representation in the graphical calculus of surface diagrams, which we present. Comment: This version fixes minor errors and adds tetrahedron coherence for braiding, as well as Yang-Baxterator coherence for syllepsis |
| Databáze: | arXiv |
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