If $X$ is aspherical, we know that $H^*(X,k) = \text{Ext}_R(k,k)$, with $R = k\pi_1$. For non-aspherical spaces, do we ever have $H^*(X,k) = \text{Ext}_R(k,k)$ for some ring $R$? Obviously we need this algebra to be graded-commutative, so perhaps we want $R$ to be a Hopf algebra.

Or, instead of a ring $R$, maybe we want a tensor category $\mathcal{C}$? Whatever the right level of generality is.

Thanks!

EDIT: I should probably have guessed that "whatever the right level of generality" was too bold of a claim; I'm not looking for things as fancy as derived categories of sheaves or ring spectra. So for now I'll stick to the question of when it really is $\text{Ext}_R(k,k)$ for a ring $R$. I'm open to $k$-linear categories that are "not too much more complicated than group rings", but have no way of making this precise.