(Bi)nary Relations in Set & Category Theory Notation

Question

I am new to both category & set theory. From what I have learned of set theory so far:

• Products between two(+) sets, A×B, create a "grid" of elements which completely covers all combinations of elements a ∈ A & b ∈ B (a,b) or (b,a) depending on which way round they were composed.
• Co-products between two(+) sets, A∪B, are essentially disjoint unions. Aka, all elements of sets A and B are put into a single set, C, and paired with a discriminator such that any overlaps (e.g. a=b) are still kept and distinguished.
• (Bi)nary Relations are generalisations of products and co-products in that they represent distinct pairs. However, they do not guarantee every element pair, like products do, nor even that all elements in a given parent set will be used, like co-products.

Now, my question is that products and co-products have their respective symbolic representations in category theory (A×B & A∪B respectively), but I have yet to see how to define a generic (bi)nary relation in notation beyond calling it some arbitrary name (as with any other generic set).

How do I correctly denote a set as being a binary relation between two sets, A & B, in both set theory and category theory?

Thanks in advance.

Answer 1

I have not seen a special notation for relations as objects/jointly monic spans, but I have run across a couple of notations for relations as morphisms in an allegory. P.T. Johnstone uses $A \looparrowright B$ in parts of Sketches of an Elephant (writing things like $\phi:A\looparrowright B$ to indicate that $\phi$ is some relation on $A,B$), and Paul Taylor has similarly used a symbol that is like $A\rightleftharpoons B$, except that there was no gap; the harpoons shared the same horizontal line. The latter doesn't seem to be a standard Latex symbol, though.

Neither of these are really standard, at the moment, so if you run across them in other contexts it's a good idea to check what the author intends.