Volume 16 (2009)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
Purchases
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

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

Abstract

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

Mathematical Subject Classification

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

Secondary: 18C20, 18D50

References
Publication

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

Authors
J P May
Department of Mathematics
The University of Chicago
Chicago, Illinois 60637
USA
http://www.math.uchicago.edu/~may