Let \(R\) be any commutative ring. The authors say that a map between reduced Kan complexes \(f: X\rightarrow Y\) is a \(\pi_1\)-\(R\)-equivalence when this map induces a bijection on fundamental groups \(\pi_1(f): \pi_1(X)\xrightarrow{\simeq}\pi_1(Y)\) and an isomorphism on the \(R\)-homology of universal covers \(H_*(\tilde{f},R): H_*(\tilde{X},R)\xrightarrow{\simeq} H_*(\tilde{Y},R)\). For instance, a map \(f\) is a \(\pi_1\)-\(\mathbb{Z}\)-equivalence if and only if \(f\) is a homotopy equivalence.
The authors consider the functor \(X\mapsto RX\), which, to any reduced Kan complex \(X\), associates the simplicial commutative \(R\)-coalgebra of chains on \(X\). Recall that a simplicial set \(X\) is reduced when its vertex set is reduced to a one-point set \(X_0 = *\). This assumption implies that the simplicial \(R\)-coalgebra \(RX\) is connected in the sense that its zero dimensional component \(RX_0\) is reduced to the ground ring \(R\).
The authors prove that the simplicial commutative \(R\)-coalgebra \(RX\) determines the simplicial set \(X\) up to \(\pi_1\)-\(R\)-equivalence when \(R\) is a field in the sense that reduced Kan complexes \(X\) and \(Y\) can be connected by a zigzag of \(\pi_1\)-\(R\)-equivalences if and only if the associated connected simplicial commutative \(R\)-coalgebras \(RX\) and \(RY\) can be connected by a zigzag of \(\Omega\)-quasi-isomorphisms, where an \(\Omega\)-quasi-isomorphism is a morphism of simplicial commutative \(R\)-coalgebras \(f: C\rightarrow D\) that induces a quasi-isomorphism \(\Omega N_*(f): \Omega N_*(C)\xrightarrow{\sim}\Omega N_*(D)\) on the cobar construction \(\Omega(-)\) of the normalized chains \(N_*(-)\) of our coalgebras. (We then use that the normalized chain complex of a simplicial \(R\)-coalgebra forms a differential graded \(R\)-coalgebra.)
The authors conjecture that an analogous statement holds over \(R = \mathbb{Z}\) and not only over a field. They establish that such a result holds in the special case of reduced Kan complexes whose universal cover is of finite type.
The proof of the results of the paper relies on the definition of a fundamental bialgebra, associated to any connected simplicial cocommutative coalgebra, and which, in the case of a simplicial coalgebra of chains \(RX\), is identified with the group algebra \(R[\pi_1(X)]\). The authors also define a notion of universal cover on the category of connected simplicial cocommutative \(R\)-coalgebras and they apply results of P. G. Goerss [Math. Z. 220, No. 4, 523–544 (1995; Zbl 0849.55011)] at the level of universal covers to obtain their statement.