E∞ ring spectra were defined in 1972, but the term has since acquired several
alternative meanings. The same is true of several related terms. The new
formulations are not always known to be equivalent to the old ones and even when
they are, the notion of equivalence needs discussion: Quillen equivalent categories can
be quite seriously inequivalent. Part of the confusion stems from a gap in the
modern resurgence of interest in E∞ structures. E∞ ring spaces were also
defined in 1972 and have never been redefined. They were central to the
early applications and they tie in implicitly to modern applications. We
summarize the relationships between the old notions and various new ones,
explaining what is and is not known. We take the opportunity to rework and
modernize many of the early results. New proofs and perspectives are sprinkled
throughout.
Keywords
commutative ring spectrum,
E∞ ring space, E∞ ring
spectrum, infinite loop space machine, operad, orthogonal
spectrum, recognition principle, S–algebra, symmetric
spectrum, unit spectrum
