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Filtered spaces, filtered objects

Tyler Lawson

Abstract

We introduce a operation on categories enriched in filtered spaces, whose effect is to turn categories of $E_1$-pages into categories of $E_2$-pages. This allows us to give a homotopical versions of several results that were previously implemented using $E_2$-model structures or more sophisticated machinery in higher algebra. We find that we can recover the homotopy theory of spaces from this page-turning operation on the homotopy theory of CW-complexes and filtration-shifting maps, a version of the cellular approximation theorem. In the category of filtered spectra, we show that this implements the procedure on filtered spectra sending the homotopy exact couple to its associated derived couple. Finally, we recover Pstragowski's category of synthetic spectra from applying this page-turning operation to the category of filtered modules over a spectral version of the Rees ring.

Filtered spaces, filtered objects

Abstract

We introduce a operation on categories enriched in filtered spaces, whose effect is to turn categories of -pages into categories of -pages. This allows us to give a homotopical versions of several results that were previously implemented using -model structures or more sophisticated machinery in higher algebra. We find that we can recover the homotopy theory of spaces from this page-turning operation on the homotopy theory of CW-complexes and filtration-shifting maps, a version of the cellular approximation theorem. In the category of filtered spectra, we show that this implements the procedure on filtered spectra sending the homotopy exact couple to its associated derived couple. Finally, we recover Pstragowski's category of synthetic spectra from applying this page-turning operation to the category of filtered modules over a spectral version of the Rees ring.

Paper Structure

This paper contains 62 sections, 82 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Outline
  4. Acknowledgements
  5. Filtered spaces and their successors
  6. Filtrations
  7. The successor functor
  8. Fibration properties
  9. Homotopy groups
  10. Categorical products
  11. Homotopy pullbacks
  12. Filtered hocolims
  13. Filtered holims
  14. Nearly constant spaces
  15. Model-independent description
  16. ...and 47 more sections

Key Result

Proposition 2.6

There is a natural isomorphism making the successor $(-)^\dag$ into a functor right adjoint to $\mathop{\mathrm{sk}}\nolimits$.

Theorems & Definitions (209)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 199 more