The Adams spectral sequence is available in any triangulated category, a general setup that has found various applications. I will mention some applications that may be relevant to toric topology. When studying the triangulated Adams spectral sequence, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution.

Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles.