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.



When is the cohomology of a space Ext(k,k)?
Yes, for any connected X we can find a $\pi$ such that $H^\*(X;k) \cong Ext\_{{k}\pi}(k,k)$


This is the Kan-Thurston theorem:


Kan, D. M.; Thurston, W. P.
Every connected space has the homology of a K(π,1). 
Topology 15 (1976), no. 3, 253–258. 


combined with the universal coefficient theorem and the isomorphism $H^\*(K(\pi,1);k) \cong Ext^\*\_{k\pi}(k,k)$


