nLab 2-bicategory

Definition

A 2-bicategory or bi-2-category is a bicategory enriched in the cartesian monoidal 2-category 2Cat of strict 2-categories. A 2-bicategory is thus a kind of semistrict tricategory.

By Corollary 8.4 of GPS 1995, every 2-bicategory is triequivalent (by a 3-functor which is locally a 2-functor) to a strict 3-category.

Examples

• Categories enriched on two sides (see KLSS02).
• Polynomials in Cat (see Weber 2015).

References

The terminology bi-2-category appears in:

• Max Kelly, A. Labella, V. Schmitt, Ross Street, Categories enriched on two sides, Journal of Pure and Applied Algebra 168 1 (2002) 53-98 [doi:10.1016/S0022-4049(01)00048-2]

The terminology 2-bicategory appears in:

• Mark Weber, Polynomials in categories with pullbacks, Theory and Applications of Categories, 30:16 (2015) 533–598. (journal)

and appears to be used more frequently in the literature.

The strictification theorem appears in:

• Robert Gordon, A. John Power, Ross Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558 (ISBN:978-1-4704-0137-5)