# (Bi)nary Relations in Set & Category Theory Notation

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?

• Given that there are lots of different relations between any two sets, how could you expect to refer to a generic one other than by calling it some arbitrary name? – Eric Wofsey Jul 12 '17 at 20:17
• @EricWofsey For instance via the use of subscripts and possibly some symbol I have yet to learn about? As it feels like a prevalent type of set I figured I've just been looking in the wrong places for it. – user3303504 Jul 12 '17 at 20:32
• I wouldn't call them generalizations of products/coproducts by any stretch; a random relation need not be any kind of universal or weakly universal structure. They have a categorical description as subobjects of products, or as the morphisms of an allegory; but not as this purported generalization. – Malice Vidrine Jul 13 '17 at 5:27
• @MaliceVidrine Hmm, not heard of allegories before, shall be the next topic on my to-learn list. Thanks for correcting me about the generalisations statement. Without looking into the category of allegories, I am assuming that there isn't any special notation used defining such binary relations? If not then I'll select an answer saying as much. Thanks for your help. – user3303504 Jul 13 '17 at 15:16
• @user3303504 - Actually, now that I've had time to think about it, I think I have seen special notation for the morphisms of an allegory (or at least, morphisms in $\mathbf{Rel}$) before; I don't think it's standardized, but let me see if I can dig it up... – Malice Vidrine Jul 13 '17 at 16:29

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.