nLab
bicategory of relations

Idea

A bicategory of relations is a (1,2)-category which behaves like the 2-category of internal relations in a regular category. The notion is due to Carboni and Walters.

Definition

A bicategory of relations is a cartesian bicategory which is locally posetal and moreover in which for every object $X$, the diagonal $\Delta :X\to X\times X$ and codiagonal? $\nabla :X\times X\to X$ satisfy the Frobenius condition?:

$$\nabla \Delta =(1\times \Delta )(\nabla \times 1).$$\nabla\Delta = (1\times \Delta)(\nabla \times 1).

There is also the dual Frobenius condition $\nabla \Delta =(\Delta \times 1)(1\times \nabla )$, and the “separable axiom” $\Delta \nabla =1$, which as far as I can tell are not in the original paper, but appear in the blog post. Do they follow from the rest of the definition?

Of course, in the locally posetal case the definition of cartesian bicategory can be simplified. (Should say something about that.)

See also

Other attempted axiomatizations of the same idea “something that acts like the category of relations in a regular category” include:

• allegories

• 1-category equipped with relations

References

• Carboni and Walters, “Cartesian Bicategories, I”

• blog post showing that any bicategory of relations is an allegory.