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

Abstract

Abstract 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 ${\mathbb{F}}$.