Homotopy theory of spectral sequences

TL;DR

This paper develops a homotopy-theoretic framework for spectral sequences of modules over a ring, establishing a fibrant object structure and comparing it with related categories.

Contribution

It introduces a fibrant object model for spectral sequences and relates it to multicomplexes and filtered complexes.

Findings

Spectral sequences form a category with a Brown-like homotopy theory.

The structure of a partial Brown category is established for spectral sequences.

Comparison results connect spectral sequences with multicomplexes and filtered complexes.

Abstract

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.