A monoidal Dold-Kan correspondence for comodules

TL;DR

This paper establishes a Quillen equivalence between simplicial and differential graded comodules, providing new methods for computing homotopy limits and connecting rational A-theory with K-theory of comodules.

Contribution

It introduces a monoidal Dold-Kan correspondence for comodules, linking simplicial and differential graded contexts with new computational techniques.

Findings

Simplicial and differential graded comodules are Quillen equivalent.

Derived cotensor products correspond under this equivalence.

Rational A-theory of a space is equivalent to K-theory of comodules.

Abstract

We provide examples of inductive fibrant replacements in fibrantly generated model categories constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond. We deduce that the rational $A$-theory of a simply connected space $X$ is equivalent to the $K$-theory of perfect chain complexes with a $C_*(X; \mathbb{Q})$-comodule structure.