Model Category Structure on Simplicial Algebras via Dold-Kan Correspondence, Part I

Document Type : Original Article

Author

Hossein Faridian

Department of Mathematics, University of Bridgeport, Bridgeport, CT, USA

10.22067/kjm.2026.545810.3674

Abstract

This article is the first part of a two-part project whose goal is to provide 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 admits a model structure. In the present paper, we recall the necessary foundations from the theory of model categories and establish several technical lemmas required for the transfer of model structures. The principal result is a detailed and self-contained proof of the existence of a cofibrantly generated model structure on the category of connective chain complexes over an arbitrary ring. This algebraic model structure will serve, in the second part of the project, as the basis for transferring a model structure to the category of simplicial commutative algebras.

Keywords

Model category

connective chain complex

simplicial module

simplicial algebra

Subjects

• 18 Category theory, homological algebra

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