Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
Roughly, $Rel$ is the category whose objects are sets and whose morphisms are (binary) relations between sets. It becomes a 2-category (in fact, a 2-poset) by taking 2-morphisms to be inclusions of relations.
$Rel$ is a 2-poset (a category enriched in the category of posets), whose objects or $0$-cells are sets, whose morphisms or $1$-cells $X \to Y$ are relations $R \subseteq X \times Y$, and whose 2-morphisms or $2$-cells $R \to S$ are inclusions of relations. The composite $S \circ R$ of morphisms $R: X \to Y$ and $S: Y \to Z$ is defined by the usual relational composite
and the identity $1_X: X \to X$ is the equality relation, in other words the usual diagonal embedding
Another important operation on relations is taking the opposite: any relation $R: X \to Y$ induces a relation
and this operation obeys a number of obvious identities, such as $(S \circ R)^{op} = R^{op} \circ S^{op}$ and $1_X^{op} = 1_X$.
It is useful to be aware of the connections between the bicategory of relations and the bicategory of spans. Recall that a span from $X$ to $Y$ is a diagram of the form
and there is an obvious category whose objects are spans from $X$ to $Y$ and whose morphisms are morphisms between such diagrams. The terminal span from $X$ to $Y$ is
and a relation from $X$ to $Y$ is just a subobject of the terminal span, in other words an isomorphism class of monos into the terminal span.
To each span $S$ from $X$ to $Y$, there is a corresponding relation from $X$ to $Y$, defined by taking the image of the unique morphism of spans $S \to X \times Y$ between $X$ and $Y$. It may be checked that this yields a lax morphism of bicategories
More generally, given any regular category $C$, one can form a 2-category of relations $Rel(C)$ in similar fashion. The objects of $Rel(C)$ are objects of $C$, the morphisms $r: c \to d$ in $Rel(C)$ are defined to be subobjects of the terminal span from $c$ to $d$, and 2-cells $r \to s$ are subobject inclusions. To form the composite of $r \subseteq c \times d$ and $s \subseteq d \times e$, one takes the image of the unique span morphism
in the category of spans from $c$ to $e$, thus giving a mono into the terminal span from $c$ to $e$. The subobject class of this mono defines the relation
and the axioms of a regular category ensure that $Rel(C)$ is a 2-category with desirable properties. Similar to what was said above, there is again a lax morphism of bicategories
There is also a functor
that takes a morphism $f: c \to d$ to the functional relation defined by $f$, i.e., the relation defined by the subobject class of the mono
Such functional relations may also be characterized as precisely those 1-cells in $Rel(C)$ which are left adjoints; the right adjoint of $\langle 1, f \rangle$ is the opposite relation $\langle f, 1\rangle$. The unit amounts to a condition
which says that the functional relation is total, and the counit amounts to a condition
which says the functional relation is well-defined.
$Rel$ does have products and coproducts; they coincide (by self-duality) and are just disjoint unions of sets. However, otherwise $Rel$ has very few (co)limits; it doesn’t even have splittings of all idempotents. All symmetric idempotents have splittings, but the order-relation $\leq \; \subseteq \{0,1\} \times \{0,1\}$ can’t be split. It follows that it can’t have (co)equalisers.
Since $Rel$ is the category of free algebras (Kleisli category) for the powerset monad, there is, indeed, very little chance of a limit of such algebras being free again. To get decent limits, one has to move to the Eilenberg-Moore category of the powerset monad, viz., the category of complete suplattices.
As $Rel$ has weak equalizers, one can take its exact completion. This happens to be the category of complete sup-lattices and sup-preserving maps. And the tensor product on $Rel$ extends to the exact completion.
The Freyd completion of REL, which is equivalent to the category of basic pairs which appear in formal topology, has all limits exactly because REL has products and weak equalizers. The Freyd completion adds freely a strong factorization system to a(ny) category C and it has products if C has products, it has equalizers if C has weak equalizers.
If you insert the category $Rel$ into the double category $\mathrm{RRel}$ of sets, mappings and relations, one has a double category with all double limits? and colimits. For instance, the obvious cartesian product $a \times b : X \times Y \to X' \times Y'$ (resp. sum $a+b : X + Y \to X' + Y'$) of two relations is indeed a product (resp. a sum) in the double category.
Similarly, many bicategories of spans, cospans, relations, profunctors, etc. have poor (co)limits, but can be usefully embedded in weak double categories (with the same objects, “strict morphisms”, “same morphisms”, suitable double cells) that have all limits and colimits.
Writing $\mathcal{V}$ for the category of suplattices, $Rel$ is a $\mathcal{V}$-category (see enriched category). With that in mind, the parallel:
is remarkable. This was probably first noticed by Dana May Latch, in the 20th century, for the category of $\mathcal{V}$ of complete suplattices.
The matrix algebra for maps to products, from coproducts, and most especially, from coproducts to products, works just as it does in the case of additive categories, when it comes to these $\mathcal{V}$-categories.
See van Kampen colimit.
It is not hard to see that a relation $R \subseteq A \times B$ is a monomorphism $A \to B$ iff the map $\mathcal{P}A \to \mathcal{P}B$ sending a subset of A to the set of all R-relatives of its members is injective; dually for epimorphisms.
A category of correspondences is a generalization of a category of relations. The composition of relations is that of correspondences followed by (-1)-truncation.
For generalizations of $Rel$ see