Suspension of operads

Abstract

We study the suspension of operads, specifically in the ∞-categories of spectra and pointed spaces, arising in the literature in the context of Koszul duality. In the stable case, we investigate a conjectured characterising property of operadic suspension posed by Heuts–Land 2024 and extend their positive result for the nonunital E_n-operads to the E_∞-operad. We also discuss an alternative approach, to characterise operadic suspension in terms of invertibility with respect to the levelwise tensor product of operads. In the unstable case, we show that constructions of Arone–Kankaanrinta 2014 and Ching–Salvatore 2022 of a ‘sphere operad’ produce equivalent operads by point-set means. We briefly discuss to what extent the approaches for the stable case could generalise to the unstable case.