nLab double fibration

Idea

A double fibration, or fibration of double categories is the appropriate analogue of Grothendieck fibration for double categories. There are several equivalent definitions; see the references.

When the source and target are known as double categories, the adjunct “double” can be omitted, as with a double functor. Also note that unlike a double category, but like a double functor, a double fibration is not a “fibration in the category of fibrations”.

References

• Michael Lambert, Discrete double Fibrations (arXiv:2101.06734), 2021.

• Geoffrey Cruttwell, Michael Lambert, Dorette Pronk, Martin Szyld, Double Fibrations (arXiv:2205.15240), 2022.

• Benedikt Fröhlich, Lyne Moser, Yoneda lemma and representation theorem for double categories, (arXiv:2402.10640), 2024

• Nathanael Arkor, John Bourke, and Joanna Ko?, Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation, arXiv:2412.07475 (2024).

• Nathanael Arkor, Exponentiable virtual double categories and presheaves for double categories, arXiv:2508.11611 (2025).