Cartesian bicategories, introduced by Carboni and Walters, form a class of bicategories in which one can take “finite products.” Typical examples of those products include the direct products of sets in the bicategory Rel of sets and relations, and the product categories in the bicategory Prof of categories and profunctors. However, they do not coincide with the actual (bi)products in those bicategories, as those products do not possess the same universal property for relations or profunctors as for functions or functors. This discrepancy leads to the intricate formulation of cartesian bicategories.
In this talk, I will introduce cartesian equipments, a certain class of double categories introduced by Aleiferi, and explain in what sense they provide a more natural framework than cartesian bicategories and still cover most of the examples studied in terms of cartesian bicategories. I will then focus on a specific subclass of cartesian equipments that closely resemble Rel as a cartesian equipment.
This talk is based on my master’s thesis.