nLab Rel

Idea

$Rel$ is the category whose objects are sets and whose morphisms are (binary) relations between sets. It becomes a 2-category $\mathbf{Rel}$ (in fact, a 2-poset) by taking 2-morphisms to be inclusions of relations.

Definition

$Rel$ is a category whose objects or $0$-cells are sets, whose morphisms or $1$-cells $X \to Y$ are relations $R \subseteq X \times Y$, and where the composite $S \circ R$ of morphisms $R: X \to Y$ and $S: Y \to Z$ is defined as the usual relational composite:

$$\{(x, z) \in X \times Z: \exists_{y \in Y} (x, y) \in R \wedge (y, z) \in S\} \subseteq X \times Z .$$

Another important operation on relations is taking the opposite, also known as the transpose or sometimes the converse. Any relation $R: X \to Y$ induces a relation

$$R^\dagger = \{(y, x) \in Y \times X: (x, y) \in R\} \subseteq Y \times X$$

This makes $Rel$ into a dagger-category: in other words, we have $(S \circ R)^\dagger = R^\dagger \circ S^\dagger$, $1_X^\dagger = 1_X$, and $(R^{\dagger})^{\dagger} = R$.

We can enhance $Rel$ to a 2-category, which we can call $\mathbf{Rel}$ to distinguish it from the mere category, by taking 2-morphisms or $2$-cells $R \Rightarrow S$ to be inclusions of relations. Since there is at most one 2-morphism between parallel 1-morphisms, $\mathbf{Rel}$ is a 2-poset: that is, an category enriched in the category of posets.

Relations and spans

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

$$X \leftarrow S \to Y$$

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

$$X \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y$$

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 pseudofunctor

$$|-|:Span \to Rel$$

Indeed for $A \leftarrow R \to B$ and $B \leftarrow S \to C$, we have:

$$|R \times_B S| = \{(a,c) \mid \exists k \in (R \times_B S)(a,c)\} = \{(a,c) \mid \exists b \in B (\exists r \in R(a,b) \land \exists s \in S(b,c))\} = |R| \odot |S|.$$

Limits and colimits

The category $Rel$ 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 the category $Rel$ 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.

Weak equalizers and completion

As the category $Rel$ 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 $C$. The Freyd completion has products if $C$ has products, and it has equalizers if $C$ has weak equalizers.

In the double category of relations

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.

As an enriched category

Writing $\mathcal{V}$ for the category of suplattices, $Rel$ is a $\mathcal{V}$-category (see enriched category). With that in mind, the parallel:

• in additive categories with zero object, finitary (co)products are biproducts
• $\mathcal{V}$-categories with zero object, arbitrary (co)products are biproducts

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.

In spans

See van Kampen colimit.

Mono/epimorphisms

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.

2-categorical aspects

Recall that we are using $\mathbf{Rel}$ to stand for the 2-category of sets, relations and inclusions. Various facts about relations can be recast in these terms. For example, a relation $L : X \to Y$ is a function from $X$ to $Y$ iff it has a right adjoint $R : Y \to X$ in $\mathbf{Rel}$. In this case, its right adjoint equals its transpose $L^\dagger : Y \to X$.

Similarly, a monad $T : X \to X$ in the 2-category $\mathbf{Rel}$ is exactly a preorder on $X$, since the existence of a unit $i: 1_X \Rightarrow T$ says that $T$ is reflexive and the existence of the multiplication $m: T^2 \Rightarrow T$ says that $T$ is transitive. This monad has an Eilenberg-Moore object iff it is an equivalence relation, in which case the Eilenberg-Moore object is the set of equivalence classes of the relation.

Relations in a category

More generally, given any regular category $C$, one can form a 2-category of relations $\mathbf{Rel}(C)$ in similar fashion. The objects of $\mathbf{Rel}(C)$ are objects of $C$, the morphisms $r: c \to d$ in $\mathbf{Rel}(C)$ 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

$$r \times_c s \to c \times e$$

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

$$s \circ r \subseteq c \times e$$

and the axioms of a regular category ensure that $\mathbf{Rel}(C)$ is a 2-category with desirable properties. Similar to what was said above, there is again a pseudofunctor

$$Span(C) \to \mathbf{Rel}(C)$$

Indeed, the proof this was strong for $C=\bf Set$ carries over to an arbitrary regular category $C$ since it is written in regular logic.

There is also a functor

$$i: C \to \mathbf{Rel}(C)$$

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

$$\langle 1, f\rangle: c \to c \times d$$

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

$$1_c \subseteq \langle f, 1 \rangle \circ \langle 1, f \rangle$$

which says that the functional relation is total, and the counit amounts to a condition

$$\langle 1, f \rangle \circ \langle f, 1 \rangle \subseteq 1_d$$

which says the functional relation is well-defined.

Related categories

• Set

• FinRel

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

• allegory, cartesian bicategory

• bicategory of relations

• Prof

• 2-poset of partial maps

• category of monic maps

References

• Grandis, M., Pare, R. :Limits in double categories. Cahiers Topologie Géom. Différentielle Catég. 40(3), 162–220 (1999)

• Bucalo, A., Rosolini, G., Completions, comonoids, and topological spaces Annals of Pure and Applied Logic 137, 104-125 (2006)

• Credits for the section on limits go to the contributors of the following threads on the categories list: categories: Limits and colimits in Rel?, categories: Limits in REL

• Aurelio Carboni, Stefano Kasangian? and Ross Street, Bicategories of spans and relations, Journal of Pure and Applied Algebra, Vol. 33, No. 3, 1984, pp. 259-267. [doi:10.1016/0022-4049(84)90061-6]

Relations may be considered in any category with a stable proper factorisation system?, but this is no more general than considering relations in a regular category, as proven in

• Gregory Maxwell Kelly. A note on relations relative to a factorization system. Category Theory. Springer, Berlin, Heidelberg, 1991.

An axiomatization of the category of sets and relations is found in:

• Andre Kornell, Axioms for the category of sets and relations (arXiv:2302.14153)

category: category