Let's say we find some regression equation $\ell$ (best fit / linear / whatever words you need to put here) for a sample $D$, subset of population $P$. This equation/model can be thought of as a vector in a vector space, $\ell \in V_D$.
Is there a way in which the theory of regression/best-fit forces this model to actually be a vector in a chain complex of vector spaces, $\ell \in V^{\bullet}_D$ so that for any intersection of subsets of the population $D_0 \cap D_1$ we get a chain map $f_{10}: V^{\bullet}_{D_0} \to V^{\bullet}_{D_1}$ which carries $\ell_0$ to $\ell_1$?
Edit: see this question for a different attempt at phrasing.
