Arrows in the category $\bf Rel$ are binary (2-valued) relations between set objects.
Do ternary, 4-term, $n$-term and variadic (2-valued) relations form categories? (Or perhaps one category?).
It may be convenient to study categorically how binary relations relate to mutual relations, as this and this question, or to represent Helly type relations.
$n$-ary relations are mentioned in nlab, but no explicit category seems defined. Neither does the concept seem to be discussed in Freyd & Scedrov's Categories, Allegories. Did I miss it?
By analogy with graphs and hypergraphs, where the former are defined by edges between pairs of vertices, whereas the latter are defined by arbitrary subsets of vertices, it's not clear offhand how would arrows be defined even for a ternary relation $R \subset X \times Y \times Z$?
I hope I understand your question correctly. If you are referring to 3-ary relations as subsets of $A\times B\times C$ and so on for $n$-ary relations in general then these are in fact already incorporated in the category $Rel$. The category $Rel$ has a monoidal structure given by the ordinary cartesian product of sets. Thus, a ternary relation $R \subseteq A\times B\times C$ can be seen as a relation $R\subseteq (A\times B)\times C$ and thus as an arrow in $Rel$ from $A\times B$ to $C$. Similarly any $n$-are relation can be interpreted as a binary relation.
Just like $Rel$ is a dagger category (that is it admits an involution) the monoidal structure on $Rel$ turns it into a cyclic operad. So, if I understand your question correctly, all of the relations you are interested in form the cyclic operad $Rel$, which is completely defined in terms of the category $Rel$ of binary relations + its monoidal structure. I hope this helps.