Allegory (mathematics)

In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. 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 means "first do , then do ".
Definition.
An allegory is a category in which
all such that
Here, we are abbreviating using the order defined by the intersection: formula_14 means formula_15
A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphism formula_16 is a binary relation between and . Composition of morphisms is composition of relations, and the anti-involution of formula_17 is the converse relation formula_18: formula_19 if and only if formula_20. Intersection of morphisms is (set-theoretic) intersection of relations.
Regular categories and allegories.
Allegories of relations in regular categories.
In a category , a relation between objects and is a span of morphisms formula_21 that is jointly monic. Two such spans formula_22 and formula_23 are considered equivalent when there is an isomorphism between and 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 has products, a relation between and is the same thing as a monomorphism into (or an equivalence class of such). In the presence of pullbacks and a proper factorization system, one can define the composition of relations. The composition formula_24 is found by first pulling back the cospan formula_25 and then taking the jointly-monic image of the resulting span formula_26
Composition of relations will be associative if the factorization system is appropriately stable. In this case, one can consider a category , with the same objects as , but where morphisms are relations between the objects. The identity relations are the diagonals formula_27
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. The category of relations for a regular category is always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersections are intersections of subobjects, computed by pullback.
Maps in allegories, and tabulations.
A morphism in an allegory is called a map if it is entire formula_28 and deterministic formula_29 Another way of saying this is that a map is a morphism that has a right adjoint in when is considered, using the local order structure, as a 2-category. Maps in an allegory are closed under identity and composition. Thus, there is a subcategory of with the same objects but only the maps as morphisms. For a regular category , there is an isomorphism of categories formula_30 In particular, a morphism in is just an ordinary set function.
In an allegory, a morphism formula_1 is tabulated by a pair of maps formula_32 and formula_33 if formula_34 and formula_35 An allegory is called tabular if every morphism has a tabulation. For a regular category , the allegory is always tabular. On the other hand, for any tabular allegory , the category 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 , and in this setting, formula_36
Unital allegories and regular categories of maps.
A unit in an allegory is an object for which the identity is the largest morphism formula_37 and such that from every other object, there is an entire relation to . An allegory with a unit is called unital. Given a tabular allegory , the category is a regular category (it has a terminal object) if and only if is unital.
More sophisticated kinds of allegory.
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.