Subfunctors and (set-theoretic) relations

Question

If a category, $C$, is concretizable, then by definition there's a faithful functor, $U$, from $C$ to Set. Let's define $U^n:C\rightarrow$ Set thus: for any $X$ in $C$, $U^n(X):=(U(X))^n$. Now, in a paper I'm reading, the authors claim that the subfunctors of $U^n$ in some sense correspond to the (set-theoretical) $n$-ary relations. This is where my confusion lies: if $V$ is a subfunctor of $U^n$, then I understand how for any object, $X$, in $C$, $V(X)$ could be taken as a surrogate for an $n$-ary relation over $X$, but I can't get how $V$ itself could be interpreted as a relation. Can anyone help me see what I'm missing?

Answer 1

Subfunctors of $U^n$ correspond, roughly, to "$n$-ary relations which can be defined uniformly across all objects of $C$." Here is an example to make things more concrete. Suppose $C = \text{Grp}$. Then, for example, $U^2$ has a subfunctor

$$G \mapsto \{ (g, h) \in G^2 : \exists f \in G : fgf^{-1} = h \}$$

which corresponds to the relation "$g$ is conjugate to $h$." This is a relation that makes sense for any group and its truth value is preserved by group homomorphisms, and that's exactly what it means to be a subfunctor of $U^n$. More generally any relation between $n$ elements $g_1, \dots g_n$ of a group which can be described using only the group operations (together with existential quantifiers or something like that; not universal ones) defines a subfunctor of $U^n$.

In a similar but maybe simpler way, natural transformations $U^n \to U$ correspond to "$n$-ary operations which can be defined uniformly across all objects of $C$." In the case of groups this recovers all operations which can be defined using repeated group multiplication together with projections and so forth and we recover an object called the Lawvere theory of groups, from which the category of groups itself can be recovered in turn. You can see my old blog post Operations and Lawvere theories for more on this. Given such a natural operation $\eta : U^n \to U$ we get an $(n+1)$-ary relation $\{ (x_1, \dots x_{n+1}) \in X^{n+1} : \eta(x_1, \dots x_n) = x_{n+1} \}$.