Model Category Structure on Simplicial Algebras via Dold-Kan Correspondence

TL;DR

This paper provides a purely algebraic proof that the category of simplicial commutative algebras over a ring forms a model category, using Dold-Kan correspondence and Quillen-Kan transfer techniques.

Contribution

It offers a self-contained algebraic proof of Quillen's result, employing Dold-Kan correspondence and novel shuffle product analysis to transfer model structures.

Findings

Established model structure on simplicial commutative algebras

Developed shuffle product to address acyclicity issues

Extended Eilenberg-Zilber theorem in this context

Abstract

This expository article sets forth a self-contained and purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. This is accomplished by starting from the model structure on the category of connective chain complexes, transferring it to the category of simplicial modules via Dold-Kan Correspondence, and further transferring it to the category of simplicial commutative algebras through Quillen-Kan Transfer Machine. The subtlety of overcoming the acyclicity condition is addressed by introducing and studying the shuffle product of connective chain complexes, establishing a variant of Eilenberg-Zilber Theorem, and carefully scrutinizing the subtle structures under study.