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What precisely are E ring spaces and E ring spectra?

J P May

Geometry & Topology Monographs 16 (2009) 215–282

DOI: 10.2140/gtm.2009.16.215

arXiv: 0903.2813


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.


commutative ring spectrum, E ring space, E ring spectrum, infinite loop space machine, operad, orthogonal spectrum, recognition principle, S–algebra, symmetric spectrum, unit spectrum

Mathematical Subject Classification

Primary: 55P42, 55P43, 55P47, 55P48

Secondary: 18C20, 18D50


Received: 14 September 2008
Accepted: 3 February 2009
Published: 16 June 2009

J P May
Department of Mathematics
The University of Chicago
Chicago, Illinois 60637