On Model Structures Relating to Spectral Sequences

In [CELW19] Cirici, Egas Santander, Livernet and Whitehouse define model structures on filtered chain complexes and bicomplexes whose weak equivalences are the $r$-weak equivalences, i.e. isomorphisms on the $(r+1)$-pages of the associated spectral sequences. In this thesis we study and generalise these model structures. These generalisations $(f\mathcal{C})_S$ and $(b\mathcal{C})_S$ for fixed such $r$ are indexed by subsets $S$ of $\{0,1,\ldots,r\}$ containing $r$ in the former case and $0$ and $r$ in the latter and are finitely cofibrantly generated.

We show each of these model structures is a left (and right) proper, cellular and stable model category. We construct a left adjoint $\mathcal{L}$ to the product totalisation functor and show, by means of Greenlees and Shipley’s cellularization principle, that it is a Quillen equivalence for suitable indexing sets $S$. As a consequence all the model categories considered thus far have equivalent homotopy categories induced via a zig-zag of Quillen equivalences given by compositions of the $\mathcal{L}$-product totalisation, identity-identity and shift-décalage adjunctions. The model structures with $r$-weak equivalences are shown to have no left Bousfield localisation to a model structure with $(r+1)$-weak equivalences. We also derive existence of various bounded variants of the model structures $(f\mathcal{C})_S$.

We then focus on the model structures on filtered chain complexes, give a classification of their cofibrant objects and cofibrations with a boundedness restriction on their filtrations and show the $(f\mathcal{C})_S$ satisfy the unit and pushout-product axioms thereby giving monoidal model categories. Furthermore the $(f\mathcal{C})_S$ satisfy the monoid axiom of Schwede and Shipley yielding model structures on modules and algebras enhancing the homotopy theory of Halperin and Tanré on filtered differential graded algebras to a model category structure.