Total cofibres of diagrams of spectra.
Hüttemann, Thomas (2005)
The New York Journal of Mathematics [electronic only]
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Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E algebras is faithful but not full.