Eilenberg-Mac Lane spectrum
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 a symmetric monoidal -groupoid.
As a symmetric monoidal ∞-groupoid the Eilenberg–Mac Lane spectrum is the abelian group of integers (under addition)
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
that give the Eilenberg–Mac Lane spectrum
chromatic homotopy theory
Revised on November 17, 2013 01:15:55
by Urs Schreiber