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Model Structures on Infinity-Categories of Filtrations

Colin Aitken

Abstract

In 1974, Gugenheim and May showed that the cohomology $\text{Ext}_A(R,R)$ of a connected augmented algebra over a field $R$ is generated by elements with $s = 1$ under matric Massey products. In particular, this applies to the $E_2$ page of the $H\mathbb{F}_p$-based Adams spectral sequence. By studying a novel sequence of deformations of a presentably symmetric monoidal stable $\infty$-category $C$, we show that for a variety of spectral sequences coming from filtered spectra, the set of elements on the $E_2$ page surviving to the $E_k$ page is generated under matric Massey products by elements with degree $s < k.$ This work is the author's PhD thesis, completed under the supervision of Peter May.

Model Structures on Infinity-Categories of Filtrations

Abstract

In 1974, Gugenheim and May showed that the cohomology of a connected augmented algebra over a field is generated by elements with under matric Massey products. In particular, this applies to the page of the -based Adams spectral sequence. By studying a novel sequence of deformations of a presentably symmetric monoidal stable -category , we show that for a variety of spectral sequences coming from filtered spectra, the set of elements on the page surviving to the page is generated under matric Massey products by elements with degree This work is the author's PhD thesis, completed under the supervision of Peter May.
Paper Structure (31 sections, 78 theorems, 167 equations, 1 figure)

This paper contains 31 sections, 78 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Roadmap
  3. Notation and Conventions
  4. Acknowledgments
  5. Filtered objects in stable $\infty$-categories
  6. The $\infty$-category $\mathop{\mathrm{Fil}}\nolimits(\mathcal{C}).$
  7. Model structures on $\infty$-categories of filtrations
  8. Review of model $\infty$-categories
  9. The $k$-projective model structure
  10. The $k$-projective model structure is closed symmetric monoidal
  11. Mapping spaces in the derived $\infty$-category
  12. The $k$-derived category is compactly generated by spheres
  13. The spectral sequence of a filtered $\mathcal{C}$-object
  14. Construction of the spectral sequence
  15. Example: the Adams spectral sequence
  16. ...and 16 more sections

Key Result

Theorem 1

Under reasonable conditions on $\mathcal{C}$, the following are true.

Figures (1)

  • Figure :

Theorems & Definitions (168)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 158 more