A monoidal Dold-Kan correspondence for 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 $C_{\ast }(X;\mathbb{Q})$-comodule structure.

arXiv comment31 pages. Final version, to appear in Journal of Pure and Applied Algebra. This paper contains some of the results in the original version of arXiv:2006.09398. arXiv admin note: substantial text overlap with arXiv:2006.09398