Homotopy Theory of Bicomplexes

TL;DR

This paper develops two model structures for bicomplexes, one based on totalisation and the other on spectral sequence $E^2$-terms, extending results to twisted complexes.

Contribution

It introduces novel model structures for bicomplexes and extends these to twisted complexes, linking weak equivalences to spectral sequence data.

Findings

Two model structures on bicomplexes are established.

Weak equivalences are characterized by totalisation and spectral sequence $E^2$-terms.

Results are extended to twisted complexes.

Abstract

We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are detected by the $E^2$-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.