Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices that a person would make: given any finite set of options $B\subseteq X$ they would pick the element with the highest utility. Another way to model those choices would be to consider the complete preorder $\succsim$ on $X$, which is induced by $u$: $$ x\succsim y\iff u(x)\geqslant u(y) $$
The question is whether the numbers assigned by $u$ have any meaning beyond this ordinal meaning. Given any increasing function $f:\mathbb{R}\rightarrow \mathbb{R}$ the alternative utility function $v=f\circ u$ generates the same choices: there's no meaningful distinction between $u$ and $v$; they have only ordinal meaning.
Therefore, theorems that assign "cardinal meaning" to a utility function $u$, do so by adding some extra structure to the set $X$ (together with some property for the utility function $u$). Some fine examples of this are the von Neumann-Morgenstern utility representation for lotteries and Savage's subjective utility representation. In both cases, two utility functions represent the same choices if and only if they are increasing affine transformations of each other: $v=au+b$, where $0<a\in \mathbb{R}$ and $b\in \mathbb{R}$. In this sense, they have cardinal meaning.
Debreu proved a beautiful theorem which gives cardinal meaning to a utility function. The "extra structure" required is that $X$ be a product: $X=X_1 \times\ldots \times X_n$. The "extra assumption" is that the underlying complete preordering $\succsim$ of $X$ treat the factors independently: fixing any subset of coordinates, the order induced on the remaining coordinates does not depend on the values of the fixed coordinates. Debreu's theorem states that, as long as $n\geqslant 3$, we can write $$ x\succsim y\iff \sum_{i=1}^n u_i (x_i)\geqslant \sum_{i=1}^n u_i (y_i) $$ and the utility functions $\left(u_i\right)_{i=1}^n$ are identified up to affine transformations.
Debreu's Theorem (Formally)
Let $\succsim$ denote a complete preordering of $X=X_1 \times \ldots \times X_n$, where the $X_i$ are connected and separable topological spaces. This order is continuous, in the sense that the sets $\{x\in X: x\succsim x'\}$ and $\{x\in X: x'\succsim x\}$ are closed.
Given a subset $I\subset \{1,\ldots,n\}$, and a point $\left(x_i\right)_{i\in I}\in \prod_{i\notin I} X_i$, the order $\succsim$ induces an order $\succsim_{I}$ over $\prod_{i\in I}X_i$. The order $\succsim$ is said to be independent if $\succsim_I$ does not depend on the choice of the point $\left(x_i\right)_{i\notin I}\in \prod_{i\notin I} X_i$.
$X_i$ is said to be essential if, for some $\left(x_j\right)_{j\neq i}$, it is not the case that $x_i\sim_i y_i$ (meaning that the inequality goes both ways).
Theorem (Debreu 1959) If $\succsim$ is independent and at least three factors are essential, then there exist functions $u_i:X_i\rightarrow \mathbb{R}$ such that $$ x\succsim y\iff \sum_{i=1}^n u_i (x_i) \geqslant \sum_{i=1}^n u_i (y_i). $$ The functions $u_i$ are unique up to positive affine transformations.
For the proof, see the original paper (page 10), or this book (section 6.12).
Question: My goal is to understand what is it that really drives the result, especially why is it important to have at least three factors.
The topological structure seems like a red herring. It is used for two main reasons: First, so that we make sure that $\succsim$ is not so rich that there just aren't enough real numbers to represent it. Instead, we might as well just assume directly that $\succsim$ is induced by some utility function on $X$. Second, it is used for solvability, as in the intermediate value theorem. This property can also be assumed directly (as done in the book).
The main assumption is that of the product structure and the independence of $\succsim$. I do not know much about category theory, but I do know that we can define a product structure, as well as a complete preorder, purely in categorical terms. Which raises the following question:
Is there a categorical analogue of Debreu's theorem?
PS: I'm not sure this is the best way to phrase the question. I've tried making it more specific, but that was hard given my poor understanding of category theory. Suggestions are welcome.
