Notions of finiteness for localisations

A research talk I gave for zygotop in Spring 2025, on my forthcoming work. Given a central map in a presentably symmetric monoidal stable infinity category, one can build an idempotent algebra – roughly speaking, as a colimit of successive tensor powers of the map – and thus obtain a smashing localisation. From this perspective, a natural notion of finiteness arises. Namely, we call a localisation compactly central if it can be constructed in this way beginning with a central map from the unit to a compact object. There is already a notion of finiteness for localisations which requires that the acyclics be generated under colimits by compact acyclics, and being compactly central turns out to in general be a stronger condition. We classify all compactly central localisations in spectra and in p-local spectra. In particular, the localisation \(L_n^f\) is compactly central.