The simplicial coalgebra of chains under three different notions of weak equivalence

TL;DR

This paper develops and compares three model structures for simplicial sets and coalgebras under different weak equivalences, establishing a homotopically full and faithful functor in the algebraically closed case.

Contribution

It constructs new model structures on simplicial sets and coalgebras for various notions of weak equivalence, and proves a key functorial property over algebraically closed fields.

Findings

Constructed three model structures on reduced simplicial sets for any commutative ring R.

Built three model structures on connected simplicial cocommutative F-coalgebras for any field F.

Proved the chain functor is homotopically full and faithful over algebraically closed fields.

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 F, we construct three model structures on the category of connected simplicial cocommutative 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 previous work of Goerss in the context of (3)-(3'), we prove that, when F is algebraically closed, the simplicial F-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each pair of model categories. More generally, when 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 F.