A distributive lattice of model structures relating to spectral sequences

The $S$-model category structures on filtered chain complexes and bicomplexes were introduced by Cirici, Egas Santander, Livernet and Whitehouse and later generalised by this author. In this paper we show they are left proper, cellular and stable model categories. We use these properties and the Cellularization Principle of Greenlees and Shipley to show that an adjunction with right adjoint the product totalisation functor from bicomplexes to filtered chains is a Quillen equivalence. Combined with other known Quillen equivalences between filtered chains this shows these model categories all present the same homotopy category. We also construct a distributive lattice whose elements are the $S$-model categories of filtered chain complexes.