From Wikipedia:
[https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem][1]

> In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function can be represented as a superposition of continuous functions of one variable. It solved a more constrained, yet more general form of Hilbert's thirteenth problem.

> The works of Andrey Kolmogorov and Vladimir Arnold established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition. More specifically,

> $ f(\mathbf {x} )=f(x_{1},\ldots ,x_{n})=\sum _{q=0}^{2n}\Phi _{q}\left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right)$

> There are proofs with specific constructions.

> In a sense, they showed that the only true multivariate function is the sum, since every other function can be written using univariate functions and summing."

There is a specialization of this theorem that says that every **symmetric** function can be expressed in the form:

$ f(x) = f(x_{1},\ldots ,x_{n})= \rho( \sum_{m=0}^{n} \phi(x_m)) $

which is like the Kolmogorov-Arnold theorem with the $\lambda_m$'s dropped.  I encountered the latter theorem in the machine-learning literature.

I'm a beginner in learning the **Yoneda lemma**, which says:

$ [ \mathcal{A}^{\mathrm{op}} , \mathbf{Set} ] (H_A, X) \cong X(A) $ 

It looks like $X$ plays the role of $\rho$ and $A$ plays the role of "sum" and the above theorem is an application of Yoneda lemma.  But I have trouble figuring out the details.  Or am I completely off?  Thanks :)

  [1]: https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem