I'm wondering if there is any work on studying positive definite kernels on (the objects of a) category. By this I mean for a category $\mathcal{C}$, find a function
$$ K: Ob\mathcal{C} \times Ob\mathcal{C} \rightarrow \mathbb{R} $$
such that for any finite subset $\{ A_i \}_{i \in I} \subset Ob\mathcal{C}$, the matrix $K[I] = [K(A_i, A_j)]_{i, j \in I}$ is positive definite over the reals. Searching the internet for this has been tricky because of the existing notion of a kernel in category theory.
I think this would be interesting because we could then construct the Reproducing kernel Hilbert Space (RHKS) $H$, and essentially be able to think of the objects of $\mathcal{C}$ as elements of $H$.
Graph kernels are an area of study in machine learning, so there is some precedent for this kind of thing.
I know there is nothing stopping me from studying such a thing, but I didn't know if there was any literature on the subject. Or is there maybe a theorem showing that any such $K$ are trivial, or none exist for certain classes of categories?
For an example, I think we can form a positive definite kernel if we have a category $\mathcal{C}$ with a finite (and nonzero) number of objects. For $A, B \in Ob\mathcal{C}$, set
$$ K(A, B) = \max\{ \#Ob\mathcal{C'}/\#Ob\mathcal{C} \mid \mathcal{C'} \text{ is a full subcategory of } \mathcal{C} \text{ containing } A \text{ and } B \text{ s.t. } Hom_{\mathcal{C'}}(A, -) \simeq Hom_{\mathcal{C'}}(B, -) \} $$
where the $\simeq$ here denotes a natural isomorphism between functors. The we have that $0 \leq K(A, B) \leq 1 \hspace{3pt} \forall A, B \in Ob\mathcal{C}$, and $K(A, B) = 1 \iff A \simeq B$ in $\mathcal{C}$. Thus if we have a subset $\{ A_i \}_{i \in I} \subset Ob\mathcal{C}$, then $K[I]$ is a real-valued symmetric matrix, and thus has real eigenvalues. For a fixed $i \in I$, we have that $0 \leq \sum_{j \neq i} |K(A_i, A_j)| \leq \#I - 1$. Then by the Gershgorin circle theorem, the eigenvalues of $K[I]$ lie between $2 - \#I$ and $\#I$. Then define the function
$$ \tilde{K}(A, B) = \begin{cases} \#Ob\mathcal{C} \text{ if } A \simeq B \\ K(A, B) \text{ otherwise}. \end{cases} $$
The eigenvalues of $\tilde{K}[I]$ are (again by Gershgorin circle theorem) between $\#Ob\mathcal{C} - \#I + 1$ and $\#Ob\mathcal{C} + \#I -1$, making it a positive definite matrix, and $\tilde{K}$ a positive definite kernel.
The function $\tilde{K}$ is contrived and probably extremely difficult to compute. But are there other similar constructions? Or assumptions/structures on the category $\mathcal{C}$ that allow more practical kernels (e.g. Abelian, monoidal, enriched)? Or is the study of kernels on categories better handled as a case by case scenario?
