Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions.
In this article we adopt the convention that morphisms compose from right to left, so RS means "first do S, then do R".
An allegory is a category in which all such that Here, we are abbreviating using the order defined by the intersection:
A first example of an allegory is the category of sets and relations.
The objects of this allegory are sets, and a morphism
In a category C, a relation between objects X and Y is a span of morphisms
are considered equivalent when there is an isomorphism between S and T that make everything commute; strictly speaking, relations are only defined up to equivalence (one may formalise this either by using equivalence classes or by using bicategories).
If the category C has products, a relation between X and Y is the same thing as a monomorphism into X × Y (or an equivalence class of such).
In the presence of pullbacks and a proper factorization system, one can define the composition of relations.
is found by first pulling back the cospan
and then taking the jointly-monic image of the resulting span
Composition of relations will be associative if the factorization system is appropriately stable.
A regular category (a category with finite limits and images in which covers are stable under pullback) has a stable regular epi/mono factorization system.
Anti-involution is defined by turning the source/target of the relation around, and intersections are intersections of subobjects, computed by pullback.
A morphism R in an allegory A is called a map if it is entire
Another way of saying this is that a map is a morphism that has a right adjoint in A when A is considered, using the local order structure, as a 2-category.
Maps in an allegory are closed under identity and composition.
An allegory is called tabular if every morphism has a tabulation.
For a regular category C, the allegory Rel(C) is always tabular.
On the other hand, for any tabular allegory A, the category Map(A) of maps is a locally regular category: it has pullbacks, equalizers, and images that are stable under pullback.
This is enough to study relations in Map(A), and in this setting,
A unit in an allegory is an object U for which the identity is the largest morphism
Given a tabular allegory A, the category Map(A) is a regular category (it has a terminal object) if and only if A is unital.
Additional properties of allegories can be axiomatized.
Distributive allegories have a union-like operation that is suitably well-behaved, and division allegories have a generalization of the division operation of relation algebra.
Power allegories are distributive division allegories with additional powerset-like structure.
The connection between allegories and regular categories can be developed into a connection between power allegories and toposes.