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The normalized singular chains of a path connected pointed space $X$ may beconsidered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with theproperty that the $0^{\text{th}}$ homology of its cobar construction, which isnaturally a cocommutative bialgebra, has an antipode, i.e. it is acocommutative Hopf algebra. We prove that a continuous map of path connectedpointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if$\mathbf{C}_*(f): \mathbf{C}_*(X)\to \mathbf{C}_*(Y)$ is an$\mathbf{\Omega}$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebrasafter applying the cobar functor $\mathbf{\Omega}$ to the underlying dgcoassociative coalgebras. The proof is based on combining a classical theoremof Whitehead together with the observation that the fundamental group functorand the data of a local system over a space may be described functorially fromthe algebraic structure of the singular chains.

The functor of singular chains detects weak homotopy equivalences