Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A monoidal Dold-Kan correspondence for comodules

Maximilien Péroux

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.

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 -theory of a simply connected space is equivalent to the -theory of perfect chain complexes with a -comodule structure.

Paper Structure

This paper contains 17 sections, 36 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Postnikov Presentations of Model Categories
  3. Postnikov presentation
  4. Fibrantly generated model categories
  5. Left-induced model categories
  6. Weak opmonoidal Quillen pair
  7. Example: Unbounded Chain Complexes
  8. The generating fibrations
  9. A Postnikov presentation
  10. Example: Simplicial and Connective Comodules
  11. Left-induced model structures on comodules
  12. Postnikov presentations
  13. A fibrant-compatible monoidal structure
  14. Application: Dold-Kan Correspondence For Comodules
  15. Lifting the Dold-Kan correspondence
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Bbbk$ be a commutative ring with global dimension zero. Let $C$ be a simply connected cocommutative differential graded coalgebra over $\Bbbk$. Then there is a Quillen equivalence: \begin{tikzcd}[column sep=large] \comod_C(\chfo) \ar[shift left=2]{r}{\Gamma} & \ar[shift left=2]{l}{\nn^C}[swap]

Theorems & Definitions (94)

  • Theorem 1.1: Theorem \ref{['thm: dold-kan correpsondance for Comodules']}, Theorem \ref{['thm: comonoidal Dold-Kan comomules eq']}
  • Theorem 1.2: Corollary \ref{['cor: rational A-theory']}
  • Definition 2.1: hess1
  • Proposition 2.2: left1
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • ...and 84 more