Model Category Structure on Simplicial Algebras via Dold-Kan Correspondence

2024/05/02 by Hossein Faridian, Faridian, Hossein
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2405.01752

openalex publication_date 2024/05/02 · openalex created_date 2024/05/11 · openalex updated_date 2026/07/28

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.

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