I will explain a new point of view on the subject of homological stability inspired by aspects of chromatic stable homotopy theory. In this worldview, the usual stabilisation map plays the role of multiplication by p on the p-local sphere spectrum S_(p), and describing the stable homology (i.e., inverting this map) therefore plays the role of calculating the rational stable homotopy groups of spheres. The "secondary stabilisation maps" described by Galatius, Kupers, and I play the role of a v_1-self map on S_(p)/p, and describing the "secondary stable homology" plays the role of calculating the v_1-periodic stable homotopy groups of spheres. Adopting this point of view, several ideas from stable homotopy theory (finite localisations, Smith--Toda complexes, Adams periodicity, ...) can be applied to obtain a good qualitative understanding of what "higher order homological stability" should be about.

(The relation with chromatic stable homotopy theory is mainly an analogy: no particular knowledge of that subject will be necessary to follow these lectures.)