In mathematics, the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

Rel has the class of sets as

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

and binary relations as

morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

. A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so . The composition of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C'' is given by :(''a'', ''c'') ∈ ''S'' o ''R'' ⇔ for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''. Rel has also been called the "category of correspondences of sets".

Properties

The category Rel has the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...

Set as a (wide) subcategory, where the arrow in Set corresponds to the relation defined by .This category is called Set_Rel by Rydeheard and Burstall. A morphism in Rel is a relation, and the corresponding morphism in the

opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...

to Rel has arrows reversed, so it is the

converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&n ...

. Thus Rel contains its opposite and is self-dual. The

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

represented by taking the converse relation provides the dagger to make Rel a dagger category. The category has two

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

s into itself given by the

hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...

: A binary relation ''R'' ⊆ ''A'' × ''B'' and its transpose ''R''^T ⊆ ''B'' × ''A'' may be composed either as ''R R''^T or as ''R''^T ''R''. The first composition results in a

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

on ''A'' and the second is on ''B''. Since the images of these hom functors are in Rel itself, in this case hom is an

internal hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory ...

. With its internal hom functor, Rel is a

closed category In category theory, a branch of mathematics, a closed category is a special kind of category. In a locally small category, the ''external hom'' (''x'', ''y'') maps a pair of objects to a set of morphisms. So in the category of sets, this is an obje ...

, and furthermore a

dagger compact category In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from thei ...

. The category Rel can be obtained from the category Set as the

Kleisli category In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise fr ...

for the

monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...

whose functor corresponds to

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

, interpreted as a covariant functor. Perhaps a bit surprising at first sight is the fact that

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

in Rel is given by the

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...

(rather than the cartesian product as it is in Set), and so is the

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...

. Rel is monoidal closed, with both the monoidal product ''A'' ⊗ ''B'' and the

internal hom In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory ...

''A'' ⇒ ''B'' given by cartesian product of sets. The category Rel was the prototype for the algebraic structure called an allegory by Peter J. Freyd and Andre Scedrov in 1990. Starting with a

regular category In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture many properties of abelia ...

and a functor ''F'': ''A'' → ''B'', they note properties of the induced functor Rel(''A,B'') → Rel(''FA, FB''). For instance, it preserves composition, conversion, and intersection. Such properties are then used to provide axioms for an allegory.

Relations as objects

David Rydeheard and Rod Burstall consider Rel to have objects that are homogeneous relations. For example, ''A'' is a set and ''R'' ⊆ ''A'' × ''A'' is a binary relation on ''A''. The morphisms of this category are functions between sets that preserve a relation: Say ''S'' ⊆ ''B'' × ''B'' is a second relation and ''f'': ''A'' → ''B'' is a function such that

xRy \implies f(x)Sf(y),

then ''f'' is a morphism. The same idea is advanced by Adamek, Herrlich and Strecker, where they designate the objects (''A, R'') and (''B, S''), set and relation.

Notes

References

* {{refend Binary relations Monoidal categories