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 term ‘double bicategory’ may be confusing, since while a double category is a category internal to $Cat$, a double bicategory is not the fully general sort of bicategory internal to $Bicat$. This issue is addressed in Morton’s work.

A double bicategory can also be regarded as a special sort of intercategory.

References

Dominic Verity, Enriched categories, internal categories and change of base Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 (TAC)

Jeffrey C. Morton, Double bicategories and double cospans Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 389-428,