# Relations as arrows and as objects - what are the arrows in the latter?

Since a relation $R$ from $X$ to $Y$ defined as a subset of $X \times Y$, the category of sets and relations is just that: the objects are sets and the arrows are the relations.

Is there a generally accepted definition of a category where (at least binary, 2-valued) relations are objects? If so, what are the morphisms between them?

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Categories, like all mathematical objects, are tools. How you define a category depends on what use you want to put it to. What use do you want to put such a category to? – Qiaochu Yuan Feb 26 '13 at 6:03
Yet many if not most mathematical structures are precisely defined, there's not a lot of options in defining what a group, ring, field, vector space, dynamical system, etc, are, correct? Why should it be different for relation - because of their diversity? – alancalvitti Feb 26 '13 at 6:04

Okay, here's one choice. $\text{Rel}$ can actually be made into a 2-category in the following way: if $R, S$ are two relations $X \to Y$, then there is a unique $2$-morphism $R \to S$ if and only if $R \subseteq S$ as subsets of $X \times Y$. If you think of relations as logical conditions, then this notion of morphism corresponds to entailment. This exhibits $\text{Rel}$ as a category enriched over posets. (Actually it is slightly better, namely it is enriched over suplattices and in particular also enriched over commutative monoids. This is closely related to the fact that $\text{Rel}$ has finite biproducts; see this blog post for details.)
Here's another choice. If $C$ is any category and you want to turn its arrows into objects of a category, you can take its arrow category $\text{Arr}(C)$. This is the category whose objects are the arrows of $C$ and whose morphisms are commutative squares. So you can also take $\text{Arr}(\text{Rel})$.
+1 for the options. It turns at least the category of endo-relations (ie, relations on a set $X$) has objects $(X,\rho)$ and arrows are relation-preserving maps, ie $f:X \to Y$, is an arrow iff $x_1 \rho x_2 \implies f(x_1) \sigma f(x_2)$. (eg, this is exactly how Sossinsky defines the category of tolerance spaces) However, how to generalize to arbitrary relations or at least binary relations in $X \times Y$, given the answers you gave above? – alancalvitti Mar 16 '13 at 1:42