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A distributive lattice of model structures relating to spectral sequences

James A. Brotherston

Abstract

The $S$-model category structures on filtered chain complexes and bicomplexes were introduced by Cirici, Egas Santander, Livernet and Whitehouse and later generalised by this author. In this paper we show they are left proper, cellular and stable model categories. We use these properties and the Cellularization Principle of Greenlees and Shipley to show that an adjunction with right adjoint the product totalisation functor from bicomplexes to filtered chains is a Quillen equivalence. Combined with other known Quillen equivalences between filtered chains this shows these model categories all present the same homotopy category. We also construct a distributive lattice whose elements are the $S$-model categories of filtered chain complexes.

A distributive lattice of model structures relating to spectral sequences

Abstract

The -model category structures on filtered chain complexes and bicomplexes were introduced by Cirici, Egas Santander, Livernet and Whitehouse and later generalised by this author. In this paper we show they are left proper, cellular and stable model categories. We use these properties and the Cellularization Principle of Greenlees and Shipley to show that an adjunction with right adjoint the product totalisation functor from bicomplexes to filtered chains is a Quillen equivalence. Combined with other known Quillen equivalences between filtered chains this shows these model categories all present the same homotopy category. We also construct a distributive lattice whose elements are the -model categories of filtered chain complexes.
Paper Structure (28 sections, 54 theorems, 29 equations, 4 figures)

This paper contains 28 sections, 54 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Filtered chain complexes
  4. Shift-décalage adjunction
  5. Bicomplexes
  6. Spectral sequences and $r$-cones
  7. Model categories
  8. Bousfield localisations
  9. Greenlees and Shipley's Cellularization Principle
  10. $S$-model structures on $f\mathcal{C}$ and $b\mathcal{C}$
  11. Adjoints to totalisation functors
  12. Left adjoint to the product totalisation functor
  13. Right adjoint to the coproduct totalisation functor
  14. The left adjoint applied to $s$-cycles
  15. Properness
  16. ...and 13 more sections

Key Result

Theorem 1

There is a zig-zag of Quillen equivalences between any two of the model categories $\left(f\mathcal{C}\right)_S$ and $\left(b\mathcal{C}\right)_T$ where $S$ and $T$ are any non-empty finite subsets of $\mathbb{N}$ with $0\in T$. The underlying adjunctions of the Quillen equivalences in question are

Figures (4)

  • Figure 1: The bicomplexes $\mathcal{ZW}_0(p,p+n)$ and $\mathcal{ZW}_r(p,p+n)$ for $r\geq 1$
  • Figure 2: The bicomplex $\mathcal{L}\mathcal{Z}_s(p,n)$
  • Figure 3: The poset $\mathcal{N}$
  • Figure 4: The poset of join-irreducibles of $\mathcal{N}$

Theorems & Definitions (117)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Lemma 2.1.4
  • Lemma 2.1.5
  • Definition 2.1.6
  • ...and 107 more