Filtered spaces, filtered objects

TL;DR

This paper introduces a new operation on categories enriched in filtered spaces that transforms $E_1$-pages into $E_2$-pages, enabling homotopical reinterpretations of classical results and connecting various categories in higher algebra.

Contribution

It presents a novel page-turning operation on filtered categories that recovers known homotopy theories and relates filtered spectra, synthetic spectra, and Rees ring modules.

Findings

Reconstructs the homotopy theory of spaces from filtered categories.

Implements the transition from homotopy exact to derived couples in filtered spectra.

Recovers Pstragowski's synthetic spectra via the page-turning operation.

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.