Double Category of Bridges

The double category of bridges for a Bridgeable Double Category $\mathcal{D}$ is a Double Category $\mathrm{Br}(\mathcal{D})$ with

• Objects are the objects of $\mathcal{D}$

• Tight morphisms are the tight morphisms of $\mathcal{D}$

• Loose morphisms are Bridges in $\mathcal{D}$

• Cells consists of a tight morphism $\varphi$ and a cell $\alpha$, along with a proof that the right-hand side commutes.

References

• Nathanael Arkor: “A (Virtual) Double Category Theorist’s Perspective on Polynomials,” 2025. https://www.youtube.com/watch?v=tCbRfjv6JQ4.