In mathematics, the category **Rel** has the class of sets as objects and binary relations as morphisms.

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]}

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 functors into itself given by the hom functor: 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 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. With its internal hom functor, **Rel** is a closed category, and furthermore a dagger compact category.

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 monoidal closed, if one defines both the monoidal product *A* ⊗ *B* and the internal hom *A* ⇒ *B* by the cartesian product of sets. It is also a monoidal category if one defines the monoidal product by the disjoint union of sets.^{[5]}

The category **Rel** was the prototype for the algebraic structure called an allegory by Peter J. Freyd and Andre Scedrov in 1990.^{[6]} Starting with a regular category 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.

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]}

**^**This category is called**Set**_{Rel}by Rydeheard and Burstall.

**^**Mac Lane, S. (1988).*Categories for the Working Mathematician*(1st ed.). Springer. p. 26. ISBN 0-387-90035-7.**^**Pareigis, Bodo (1970).*Categories and Functors*. Pure and Applied Mathematics. Vol. 39. Academic Press. p. 6. ISBN 978-0-12-545150-5.**^**Bergman, George (1998). "§7.2 RelSet".*An Invitation to General Algebra and Universal Constructions*. Henry Helson. ISBN 0-9655211-4-1.- ^
^{a}^{b}Barr, Michael; Wells, Charles (1990).*Category Theory for Computing Science*(PDF). Prentice Hall. p. 181. ISBN 978-0131204867. **^**Fong, Brendan; David I Spivak (2019). "Supplying bells and whistles in symmetric monoidal categories". arXiv:1908.02633 [math.CT].**^**Freyd, Peter J.; Scedrov, Andre (1990).*Categories, Allegories*. North Holland. pp. 79, 196. ISBN 0-444-70368-3.**^**Rydeheard, David; Burstall, Rod (1988).*Computational Category Theory*. Prentice-Hall. p. 41. 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. Cambridge University Press. p. 115. ISBN 978-0-521-44179-7.