  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unpublished result of M. J. Hopkins that certain towers of generalized Moore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 K(n)-local sphere, are E-infinity 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.
