symmetric monoidal (∞,1)-category of spectra
A symmetric ring spectrum is a ring spectrum modeled as a symmetric spectrum. Equivalently this is a monoid object with respect to the symmetric monoidal smash product of spectra, for symmetric spectra. Morever, if we regard symmetric spectra as -modules, as discussed at Model categories of diagram spectra, then this, in turn, is equivalent to a -algebra, where is the standard model of the sphere spectrum as a symmetric spectrum.
There is a model structure for symmetric ring spectra (MMSS 00) under which symmetric ring spectra represent genuine A-infinity rings, and commutative symmetric ring spectra represent genuine E-infinity rings. This fact is one of the key motivation for passing from sequential spectra to the richer model of symmetric spectra (and possibly further to other highly structured spectra such as orthogonal spectra and excisive functors).
Despite all this, the component expression of the structure in a symmetric ring spectrum, in the fashion of functors with smash product, is rather straightforward, and directly analogous to the structure in a dg-algebra: essentially there is for all pairs of indices a pairing between the component spaces of the spectrum in these degrees
such that this respects the given action of the symmetric groups and of suspensions, and such that it is is associative and unital in the evident way.
Write for the category of pointed compactly generated topological spaces.
One minimal way to state the definition is as follows. We follow conventions as used at model structure on orthogonal spectra.
A commutative symmetric ring spectrum is
a sequence of component spaces for ;
a basepoint preserving continuous left action of the symmetric group on ;
for all a multiplication map
(a morphism in )
two unit maps
such that
intertwines the -action;
(associativity) for all the following diagram commutes (where we are notationally suppressing the associators of )
(unitality) for all the following diagram commutes
and
(commutativity) for all the following diagram commutes
where denotes the permutation which shuffles the first elements past the last elements.
A homomorphism of symmetric commutative ring spectra is a sequence of -equivariant pointed continuous functions such that the following diagrams commute for all
and and .
Write
for the resulting category of symmetric commutative ring spectra.
There is an analogous definition of orthogonal ring spectrum and we write
for the category that these form.
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)
Stefan Schwede, Symmetric spectra, 2012 (pdf)
The structure of a symmetric ring spectrum on KO is discussed in
-theory_, Topology 40 (2001) 299-308
Last revised on June 15, 2017 at 13:38:18. See the history of this page for a list of all contributions to it.