Classical homological stability from the point of view of cells

We explain how to interpret the complexes arising in the "classical" homology stability argument (eg in the framework of Randal -Williams and Wahl) in terms of higher algebra, which leads to a new proof of homological stability in this setting. The key ingredient is a theorem of Damiolini on the contractibility of certain arc complexes. We also explain how to directly compare the connectivities of these complexes with that of the "splitting complexes" of Galatius, Kupers and Randal -Williams.