Category of relations
In
A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A × B.
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.[1]
Rel has also been called the "category of correspondences of sets".[2]
Properties
The category Rel has the category of sets Set as a (wide) subcategory, where the arrow f : X → Y in Set corresponds to the relation F ⊆ X × Y defined by (x, y) ∈ F ⇔ f(x) = y.[note 1][3]
A morphism in Rel is a relation, and the corresponding morphism in the opposite category to Rel has arrows reversed, so it is the converse relation. Thus Rel contains its opposite and is self-dual.[4]
The involution represented by taking the converse relation provides the dagger to make Rel a dagger category.
The category has two
The category Rel can be obtained from the category Set as the Kleisli category for the monad whose functor corresponds to power set, interpreted as a covariant functor.
Perhaps a bit surprising at first sight is the fact that product in Rel is given by the disjoint union[4]: 181 (rather than the cartesian product as it is in Set), and so is the coproduct.
Rel is
The category Rel was the prototype for the algebraic structure called an
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 then f is a morphism.[7]
The same idea is advanced by Adamek, Herrlich and Strecker, where they designate the objects (A, R) and (B, S), set and relation.[8]
Notes
- ^ This category is called SetRel by Rydeheard and Burstall.
References
- ISBN 0-387-90035-7.
- ISBN 978-0-12-545150-5.
- ISBN 0-9655211-4-1.
- ^ ISBN 978-0131204867.
- arXiv:1908.02633 [math.CT].
- ISBN 0-444-70368-3.
- ISBN 978-0131627369.
- ^ Adamek, Juri; Herrlich, Horst; Strecker, George E. (2004) [1990]. "§3.3, example 2(d)". Abstract and Concrete Categories (PDF). KatMAT Research group, University of Bremen. p. 22. Archived from the original (PDF) on 2022-08-11.
- Borceux, Francis (1994). Categories and Structures. Handbook of Categorical Algebra. Vol. 2. ISBN 978-0-521-44179-7.