Special and general types
Stable Homotopy theory
For an abelian group, the Eilenberg-Mac Lane spectrum is the spectrum that represents the ordinary cohomology theory/ordinary homology with coefficients in .
By default is taken to be the integers and hence “the Eilenberg-MacLane spectrum” is , representing integral cohomology.
As symmetric/orthogonal spectra
We discuss the model of Eilenberg-MacLane spectra as symmetric spectra and orthogonal spectra. To that end, notice the following model for Eilenberg-MacLane spaces.
(Aguilar-Gitler-Prieto 02, def. 6.4.20)
For a countable abelian group, then the reduced -linearization (def. 1) is an Eilenberg-MacLane space, in that its homotopy groups are
(in particular for then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).
(Aguilar-Gitler-Prieto 02, corollary 6.4.23)
For a countable abelian group, then the orthogonal spectrum incarnation of the Eilenberg-MacLane spectrum is the orthogonal spectrum with
being the reduced -linearization (def. 1) of the representation sphere ;
hence for then
-action on induced from the canonical -action on (representation sphere);
The incarnation of as a symmetric spectrum is the same, with the group action of replaced by the subgroup action of the symmetric group .
If is a commutative ring, then the Eilenberg-MacLane spectrum becomes a commutative orthogonal ring spectrum (or symmetric ring spectrum, respectively) by
taking the multiplication
to be given by
taking the unit maps
to be given by the canonical inclusion of generators
(Schwede 12, example I.1.14, Schwede 15, V, costruction 3.21)
As symmetric monoidal -groupoids
Under the identification of connective spectra with “abelian infinity-groups” the Eilenberg-MacLane spectrum simply is the group .
Here the set is regarded as a discrete groupoid (one object per integer, no nontrivial morphisms) whose symmetric monoidal structure is that given by the additive group structure on the integers.
Accordingly, the infinite tower of suspensions induced by this is the sequence of ∞-groupoids
that in this case happen to be strict omega-groupoids. The strict omega-groupoid has only identity -morphisms for all , except for , where are the endomorphisms of the unique identity -morphism.
The strict ∞-groupoid is the one given under the Dold-Kan correspondence by the crossed complex of groupoids that is trivial everywhere and has the group in degree .
Under the Quillen equivalence
between infinity-groupoids and topological spaces (see homotopy hypothesis) this sequence of suspensions of maps to the sequence of Eilenberg–Mac Lane spaces
chromatic homotopy theory
Ordinary homology spectra split
ordinary homology spectra split: For any spectrum and an Eilenberg-MacLane spectrum, then the smash product (the -ordinary homology spectrum) is non-canonically equivalent to a product of EM-spectra (hence a wedge sum of EM-spectra in the finite case).
Textbook accounts include
Lecture notes include
Peter May, chapter 22 of A concise course in algebraic topology (pdf)
John Rognes, section 3.2 3.4 of The Adams spectral sequence, 2012 (pdf)
As symmetric spectra
as orthogonal spectra: