The simplicial coalgebra of chains over a field $\mathbb{F}$ determines spaces up to $\pi_1$-$\mathbb{F}$-equivalence

Let $\mathbb{F}$ be a field. A map between reduced Kan complexes is called a $\pi_1$-$\mathbb{F}$-equivalence if it induces an isomorphism on fundamental groups and an isomorphism on homology with $\mathbb{F}$-coefficients on universal covers. A map between two connected simplicial cocommutative $\mathbb{F}$-coalgebras is called an $\Omega$-quasi-isomorphism if it induces a quasi-isomorphism of differential graded (dg) algebras after applying the cobar functor to the induced map between the dg coassociative coalgebras of normalized chains. We prove that two reduced Kan complexes $X$ and $Y$ can be connected by a zig-zag of $\pi_1$-$\mathbb{F}$-equivalences if and only if their simplicial cocommutative $\mathbb{F}$-coalgebras of chains $\mathbb{F}X$ and $\mathbb{F}Y$ can be connected by a zig-zag of $\Omega$-quasi-isomorphisms. To achieve this, we develop a theory of twisting cochains and twisted tensor products at the level of simplicial coalgebras and algebras, which we use to construct the universal cover of an abstract connected simplicial cocommutative coalgebra.