We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M J Hopkins that certain towers of generalized Moore spectra, closely related to the –local sphere, are –algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.
"Commutative ring objects in pro-categories and generalized Moore spectra." Geom. Topol. 18 (1) 103 - 140, 2014. https://doi.org/10.2140/gt.2014.18.103