Is there a combinatorial/topological treatment of statistical independence?

It is aimless to extend "statistical independence" beyond category of $\sigma$-subalgebras and probability-preserving morphisms on a fixed probability space $\prod,\mathcal{A}$ as pointed out by the comment by @Dima Pasechnik.

We restrict ourselves to the category of $\sigma$-subalgebras, whose objects are the collection $\Lambda$ of all $\sigma$-subalgebras of a given $\sigma$-algebra. Some generalizations can be attained using dilations [Nagel&Palm] if you really want to embed the algebra associated with random variables on $\sigma$-subalgebras into $L^p$.

In such a category, most part of statistics are treating the separable base space, therefore via Rokhlin-Maharam theorem we know that statisticians are treating only $[0,1]\times[0,1]$ up to probability-preserving mappings. Two $\sigma$-subalgebras $\mathcal{A}_1,\mathcal{A}_2$ over the same space $(\prod,\mathcal{A},P)$ are said to be independent if $P(AB)=P(A)P(B),\forall A\in \mathcal{A}_1;B\in \mathcal{A}_2$; if we resort to axiomatic information theory and define an entropy functional $H$, then equivalently we can say these two algebras are independent if $H(\mathcal{A}_1\cap\mathcal{A}_2)=H(\mathcal{A}_1)+H(\mathcal{A}_2)$.

Now consider a probability preserving automorphism $\tau$ on $\prod,\mathcal{A}$, then such an automorphism induces automorphism $\bar{\tau}$ on the lattice $\Lambda$ of all $\sigma$-subalgebras of $\mathcal{A}$. Using the entropy characterization of independence, we must stipulate $H(\mathcal{A}_1 )=H(\bar{\tau}(\mathcal{A}_1)),\forall \mathcal{A}_1\in \Lambda$. This pretty much characterized the statistical independence within the category. This the the grand framework behind [Forman] as I understood.

The problem is that the structure of the dual notion $\Lambda$ of the $\sigma$-subalgebras remains mysterious as far as I know (If you do not even know the structure of this fundamental dual notion there is little hope that you can apply any topological trick). Thus a fuller treatment of the notion of independence is not very realistic unless we got a fuller knowledge of the structure of $\Lambda$, say a classification of all such subalgebras up to probability preserving transformations. There is one [MO] post asked earlier by me which contains some interesting links treating independence from categorical viewpoint. After I gone through it, as you can probably feel, the development is still very primitive.

[Nagel&Palm]Nagel, R. A. I. N. E. R., and Günther Palm. "Lattice dilations of positive contractions on L p-spaces." Canad. Math. Bull 25 (1982): 371-374.

[Forman]Forman, Robin. "Morse theory and evasiveness." Combinatorica 20.4 (2000): 489-504.

[MO]What is the algebraic equivalent of independent elements?

This is an old question which already has an accepted answer, but I believe the theory of so-called 'gaussoids' is precisely a combinatorial abstraction of the idea of independence in probability (see e.g. https://arxiv.org/abs/1710.07175).