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Adams' cobar construction as a monoidal $E_{\infty}$-coalgebra model of the based loop space

Anibal M. Medina-Mardones, Manuel Rivera

Abstract

We prove that the classical map comparing Adams' cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty$-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams' map preserves monoidal coalgebra structures.

Adams' cobar construction as a monoidal $E_{\infty}$-coalgebra model of the based loop space

Abstract

We prove that the classical map comparing Adams' cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal -coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams' map preserves monoidal coalgebra structures.

Paper Structure

This paper contains 26 sections, 5 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conventions and preliminaries
  3. Coalgebras
  4. Monoids
  5. Simplicial theory
  6. Cubical theory
  7. Adams's model of the based loop space
  8. The based loop space
  9. The cobar construction
  10. Adams's map
  11. An explicit choice
  12. Necklaces
  13. From necklaces to cubes
  14. Cubical cobar construction
  15. Relation to the cobar construction
  16. ...and 11 more sections

Key Result

Theorem 1

The following diagram commutes up to natural isomorphisms: \begin{tikzcd} [row sep=small] & \Mon_{\coAlg_\UM} \arrow[d] \\ \Mon_{\cSet} \arrow[ru, "\cchainsUM", out=70, in=180, near start] \arrow[r, "\cchainsA"] & \Mon_{\coAlg} \arrow[d] \\ \sSet^0 \arrow[r, "\cobar \schainsA"] \arrow[u, "\c

Figures (3)

  • Figure 1: In red, the path $\theta_3(\mathbf{t}) \in P(\mathbb{\Delta}^3, v_0, v_3)$ associated to a $\mathbf{t} \in \mathbb{I}^2$.
  • Figure 2: The faces of the $2$-cube in blue, labeled by a red element in their corresponding family of paths.
  • Figure 3: In red, the path $\theta_{(2,1,3,2)}(\mathbf{t})$ in $P(\mathbb{\Delta}^2 \vee \mathbb{\Delta}^1 \vee \mathbb{\Delta}^3 \vee \mathbb{\Delta}^2)$ associated to an element $\mathbf{t}$ in $\mathbb{I}^{4}$.

Theorems & Definitions (13)

  • Theorem
  • Theorem
  • Remark \oldthetheorem
  • Remark
  • Remark
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Remark
  • Lemma \oldthetheorem
  • ...and 3 more