The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence.

We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring |$R$|⁠. The weak equivalences are given by: (1) an |$R$| -linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an |$R$| -homology equivalence between universal covers, and (3) |$R$| -homology equivalences. Analogously, for any field |${\mathbb{F}}$|⁠ , we construct three model structures on the category of connected simplicial cocommutative |${\mathbb{F}}$| -coalgebras. The weak equivalences in this context are (1 ′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2 ′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3 ′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3 ′), we prove that, when |${\mathbb{F}}$| is algebraically closed, the simplicial |${\mathbb{F}}$| -coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when |${\mathbb{F}}$| is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of |${\mathbb{F}}$|⁠. [ABSTRACT FROM AUTHOR]