"Het" ternary and n-ary relations in $\bf Rel$?

Question

Are n-ary relations, with n > 2, in $\bf Rel$ - the category of sets as objects and relations as arrows?

By "het" (heterogeneous) relations I mean relations between distinct sets, so $X \to Y$, as opposed to "hom" (homogeneous) relations $X \to X$.

Any relation, regardless of arity, can be expressed as a subset of product of sets, so a het ternary relation is a subset $\rho \subset X \times Y \times Z$.

Are such relations in $\bf Rel$? As a starting point, $\bf Set$ has products but how is $\rho$ represented as an arrow? If $\rho : X \times Y \to Z$ why is $X \times Y$ in $dom$ and $Z$ in $cod$?

Certainly any permutation should represent the same $\rho$ but then how is the equality between the permutations encoded in $\bf Rel$?

Answer 1

The arrows of $\mathbf{Rel}$ are only binary relations. So no, there aren't any ternary relations in $\mathbf{Rel}$.

(of course, $\mathbf{Rel}$ does have binary relations from $X \times Y$ to $Z$)

"Hom", incidentally, is a very poor choice of notation here, given that the trigraph already has a meaning (and refers to a rather fundamental notion of category theory).

Answer 2

$\text{Rel}$ encodes relations in various ways; for example, a relation $X \to Y$ can be encoded equivalently as

• a morphism $X \to Y$,
• a morphism $X \times Y \to 1$, or
• a morphism $1 \to X \times Y$.

Here $X \times Y$ refers to the set-theoretic product, which is not the categorical product in $\text{Rel}$; it is just a monoidal product turning $\text{Rel}$ into a closed monoidal category. Using this monoidal product, you can encode $n$-ary relations, again in various ways. For example, a ternary relation among sets $X, Y, Z$ can be encoded equivalently as

• a morphism $X \times Y \to Z$,
• a morphism $Y \to X \times Z$,
• a morphism $1 \to X \times Y \times Z$,

and so forth. This is closely analogous to how a linear transformation between finite-dimensional vector spaces $U \to V$ can be encoded equivalently as

• a morphism $U \to V$,
• a morphism $1 \to U^{\ast} \otimes V$,
• a morphism $U \otimes V^{\ast} \to 1$.

The precise structural similarity is that $\text{Rel}$ and $\text{FinVect}$ are both dagger compact, but $\text{Rel}$ has the additional distinction that every object is self-dual.