An allegory is a category with properties meant to reflect the properties one expects of a category of relations. The notion was first introduced in the book Categories, Allegories by Freyd and Scedrov.
An allegory is a (1,2)-category equipped with an involution which is the identity on objects, such that
A map in an allegory is a morphism that has a right adjoint.
A tabulation of a morphism is a pair of maps such that and . An allegory is tabular if every morphism has a tabulation, and pretabular if every morphism is contained in one that has a tabulation.
Every regular category, and indeed every locally regular category, has a tabular allegory of internal binary relations. Conversely, by restricting to the morphisms with left adjoints (“maps”) in a tabular allegory, we obtain a locally regular category. These constructions are inverse, so tabular allegories are equivalent to locally regular categories.
A locally regular category has finite products if and only if its tabular allegory of relations has top elements in its hom-posets.
Finally, a unit in an allegory is an object such that is the greatest morphism , and every object admits a morphism such that . A locally regular category has a terminal object (hence is regular) if and only if its tabular allegory of relations has a unit.
Thus, regular categories are equivalent to unital tabular allegories.
Other attempted axiomatizations of the same idea “something that acts like the category of relations in a regular category” include:
bicategory of relations (a special sort of cartesian bicategory)
Categories, Allegories?
The Elephant, chapter A3.
blog post showing that any bicategory of relations is an allegory.