Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monoidal model structures on filtered chain complexes relating to spectral sequences

James A. Brotherston

Abstract

We establish monoidal model structures on model categories of filtered chain complexes constructed by Cirici, Egas Santander, Livernet and Whitehouse whose weak equivalences are the quasi-isomorphisms on the $r$-page of the associated spectral sequences. In doing so we provide a partial classification of cofibrant objects and cofibrations of the model structures involving a boundedness restriction on the filtration. As a consequence we also obtain, by results of Schwede and Shipley, cofibrantly generated model structures on the categories of filtered differential graded algebras as well as their modules.

Monoidal model structures on filtered chain complexes relating to spectral sequences

Abstract

We establish monoidal model structures on model categories of filtered chain complexes constructed by Cirici, Egas Santander, Livernet and Whitehouse whose weak equivalences are the quasi-isomorphisms on the -page of the associated spectral sequences. In doing so we provide a partial classification of cofibrant objects and cofibrations of the model structures involving a boundedness restriction on the filtration. As a consequence we also obtain, by results of Schwede and Shipley, cofibrantly generated model structures on the categories of filtered differential graded algebras as well as their modules.
Paper Structure (29 sections, 66 theorems, 50 equations, 10 figures)

This paper contains 29 sections, 66 theorems, 50 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Homotopy theories and spectral sequences in the literature
  3. The work of Cirici, Egas Santander, Livernet and Whitehouse
  4. Contents of this paper
  5. Structure of the paper
  6. Preliminaries
  7. Filtered chain complexes
  8. Spectral sequences from filtered chain complexes
  9. The shift-décalage adjunction of Deligne
  10. Monoidal model categories
  11. Muro's results
  12. Existing model structures
  13. Projective model structure on chain complexes
  14. Construction of new model structures
  15. Cofibrations
  16. ...and 14 more sections

Key Result

Theorem 1

There is a right proper cofibrantly generated model structure on $f\mathcal{C}$ denoted $\left(f\mathcal{C}\right)_S$ with generating cofibrations $I_S$ and generating acyclic cofibrations $J_S$.

Figures (10)

  • Figure 1: Components of the pushout: $\mathcal{B}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$, $\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ and $\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$. The central diagram maps via $k_1$ into the top diagram and via $k_2$ into the bottom.
  • Figure 2: Named components of the pushout corresponding to \ref{['domainPushoutProductComponents']}
  • Figure 3: Named codomain of the pushout-product $i\boxtimes j$
  • Figure 4: Representation in $\mathbb{Z}_\infty$-chains of $\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{Z}_{r+1}(q,m)$ with differentials described in \ref{['greek_i']}
  • Figure 5: Representation in $\mathbb{Z}_\infty$-chains of $\mathcal{Z}_{r+1}(p,n)\otimes\mathcal{B}_{r+1}(q,m)$ with differentials described in \ref{['greek_i']}
  • ...and 5 more figures

Theorems & Definitions (136)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Remark 2.1.4
  • Lemma 2.1.5: CELW
  • ...and 126 more