- Category of relations
In
mathematics , the category Rel has the class of sets as objects andbinary relation s as morphisms.A morphism (or arrow) "R" : "A" → "B" in this category is a relation between the sets "A" and "B", so nowrap| "R" ⊆ "A" × "B".
The composition of two relations "R": "A" → "B" and "S": "B" → "C" is given by::("a", "c") ∈ "S" o "R" if (and only if) for some "b" ∈ "B", ("a", "b") ∈ "R" and ("b", "c") ∈ "S".
Properties
Category Rel has the
category of sets Set as a (wide)subcategory , where the arrow (function) nowrap| "f" : "X" → "Y" in Set corresponds to the functional relation nowrap| "F" ⊆ "X" × "Y" defined by: nowrap|1= ("x", "y") ∈ "F" ⇔ "f"("x") = "y".Category Rel can be obtained from category Set as the
Kleisli category for the monad whosefunctor corresponds topower set , interpreted as a covariant functor.The
involutary operation of taking the inverse (or converse) of a relation, where nowrap| ("b", "a") ∈ "R"−1 : "B" → "A" if and only if nowrap| ("a", "b") ∈ "R" : "A" → "B", induces a contravariant functor nowrap| Relop → Rel that leaves the objects invariant but reverses the arrows and composition. This makes Rel into adagger category . In fact, Rel is adagger compact category .ee also
*
Allegory (category theory) . The category of relations is the paradigmatic example of an allegory.
Wikimedia Foundation. 2010.