Homology, Homotopy and Applications

Volume 26 (2024)

Number 1

Homotopy theory of spectral sequences

Pages: 69 – 86

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a5


Muriel Livernet (Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France)

Sarah Whitehouse (School of Mathematics and Statistics, University of Sheffield, England)


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.


spectral sequence, model category

2010 Mathematics Subject Classification


Received 18 July 2022

Received revised 5 January 2023

Accepted 2 February 2023

Published 21 February 2024