Is there a (non-Abelian) homology theory that realizes the following:
Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ are isomorphic.
Or maybe, the following? ("You need the map...")
Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Let $f:X\to Y$ be a continuous map which induces the "chain map" $f_*:C(X)\to C(Y)$. Then $f$ is a homotopy equivalence (it admits a $g:Y\to X$ such that $fg$ and $gf$ are homotopic to the identities) if and only if $f_*$ is an isomorphism.
If yes: which one/ones? An introduction to such theory?
If no: not yet, or is it impossible? Why?
No restriction for the objects in the complexes, they can be groups, modules, groupoids, anything else.
Of course, $X,Y$ may be required to be "nice enough" for the above to work. So, please state also the technical requirements. Moreover:
Same questions, but with cohomology instead of homology?