nLab
allegory

Idea

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.

Definition

An allegory is a (1,2)-category $A$ equipped with an involution $(-{)}^o:A^{\mathrm{op}}\to A$ which is the identity on objects, such that

• each hom-poset $A(x,y)$ has binary intersections, and
• the modular law holds: for $\varphi :x\to y$, $\psi :y\to z$, and $\chi :x\to z$, we have $\psi \varphi \cap \chi \le \psi (\varphi \cap {\psi }^o\chi )$.

See also

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)

• 1-category equipped with relations

References

• Categories, Allegories?

• The Elephant, chapter A3.

• Wikipedia.

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