Homological stability via E_k-algebras

Homological stability theorems have a long history, but for most of that time were treated as ad hoc results which happened to hold in many examples, with some informal principles for when they might be expected to hold. A more conceptual view was taken by Kupers--Miller, and extensively developed by Galatius-Kupers-Randal-Williams. In particular, a second-order form of homological stability was discovered in several examples. There is an emerging view that one should always have an entire hierarchy of (secondary, tertiary, ...) stabilisation maps, and while this is on first theoretical footing it has not yet been explored in depth in any example "from nature". The goal of this project is to do so.