Is there a way to axiomatize the category of sets and relations?

Question

The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?

Answer 1

Here's one way presented in at least one publication:

Objects are pairs $(X,\rho)$ with $X$ a set and $\rho \subset X \times X $ is a relation on $X$.

For a pair of objects $(X,\rho)$ and $(Y,\sigma)$, arrows are maps $f: X \to Y$ that preserve the relation, ie, $x_1 \rho x_2 \implies f(x_1) \sigma f(x_2)$ for $x_1, x_2 \in X$. (Are arrows set functions or relations?)

(Further, there is an induced relation between parallel pairs of arrows but I'll have to recall the details)

This is for example exactly how Schreider and Sossinsky defines the category of tolerance spaces, ie, sets with a reflexive, symmetric relation on them.

Note the relations here are not general:

(1) the relations are binary

(2) are "endo" since they're defined on the set $X$, as opposed to relating distinct sets ($\xi \subset X \times Y$)

(3) are 2-valued.

I don't know if considering arrows that reflect relation rather than preserve it is still the same category (dualized).

Also you wrote "axiomatize" so I think this answer suffices. If you has asked for basic properties such as what limits exist, that's another story...

Answer 2

From Categories, Allegories by Freyd, Scedrov:

2.414. If $\mathbf{C}$ is a topos, then $\mathscr{Rel}(\mathbf{C})$ is a power allegory. Conversely, if $\mathbf{A}$ is a unitary tabular power allegory, then $\mathscr{Map}(\mathbf{A})$ is a topos.

So a reasonable approach would be to start with "$\mathbf{A}$ unitary tabular power allegory", then translate well-pointed, NNO, and AC into the allegory language.