A double bicategory is a structure similar to a double category, but where composition of both vertical and horizontal arrows is only weakly associative and unital.
To define this notion, we need to include extra shapes of 2-dimensional cells in addition to the squares that appear in a double category: we also need vertical and horizontal globes. The reason is that if we have associators for horizontal morphisms given by squares, it appears to be impossible to formulate the pentagon identity for these squares unless either 1) composition of vertical morphisms is strict, or 2) we introduce cells with new shapes. In case 1) — that is, if associativity and the unit laws hold strictly in one direction — we have a ‘pseudo double category’, as studied by Grandis, Paré and Fiore. (See double category for more on this concept.) In case 2) we have a double bicategory.
The concept and terminology here were introduced by Verity:
The definition can also be found here:
and also here:
The term ‘double bicategory’ may be confusing, since while a double category is a category internal to , a double bicategory is not the fully general sort of bicategory internal to . This issue is addressed in Morton’s work.