This paper defines double fibrations (fibrations of double categories) and describes their key examples and properties. In particular, it shows how double fibrations relate to existing fibrational notions such as monoidal fibrations and discrete double fibrations, proves a representation theorem for double fibrations, and shows how double fibrations are a type of internal fibration.
Keywords: double categories, fibrations of categories, internal categories, internal fibrations
2020 MSC: 18N10, 18D30
Theory and Applications of Categories, Vol. 38, 2022, No. 35, pp 1326-1394.