Spectral sequences via linear presheaves

TL;DR

This paper develops a homotopy-theoretic framework for spectral sequences using linear presheaves, introducing extended spectral sequences and model structures to better understand their categorical properties.

Contribution

It introduces the category of extended spectral sequences, establishes model structures, and clarifies the concept of dcalage functors within a homotopical context.

Findings

Spectral sequences form a fibrant object in a model structure for infinity categories.

The category of extended spectral sequences is bicomplete and admits various model structures.

Dcalage functors are rigorously defined as adjoints to shift functors, clarifying their role.

Abstract

We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two d\'ecalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term d\'ecalage in connection with spectral sequences.