nLab
Eilenberg-Mac Lane spectrum

Idea

For $A$ an abelian group, the Eilenberg-Mac Lane spectrum $HA$ is the spectrum that represents the ordinary cohomology theory with coefficients in $A$.

By default $A$ is taken to be the integers and hence “the Eilenberg-MacLane spectrum” is $H\mathbb{Z}$, representing integral cohomology.

As a symmetric monoidal $\mathrm{\infty }$-groupoid.

As a symmetric monoidal ∞-groupoid the Eilenberg–Mac Lane spectrum is the abelian group of integers (under addition)

$$E_{\mathrm{EM}}:=\mathbb{Z}.$$E_{EM} := \mathbb{Z} \,.

Here the set $\mathbb{Z}$ 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

$$\mathbb{Z},B\mathbb{Z},B^2\mathbb{Z},B^3\mathbb{Z},\cdots$$\mathbb{Z}, \mathbf{B} \mathbb{Z}, \mathbf{B}^2 \mathbb{Z}, \mathbf{B}^3 \mathbb{Z}, \cdots

that in this case happen to be strict omega-groupoids. The strict omega-groupoid $B^n\mathbb{Z}$ has only identity $k$-morphisms for all $k$, except for $k=n$, where ${\mathrm{Mor}}_n(B^n\mathbb{Z})=\mathbb{Z}$ are the endomorphisms of the unique identity $(n-1)$-morphism.

The strict ∞-groupoid $B^n\mathbb{Z}$ is the one given under the Dold-Kan correspondence by the crossed complex of groupoids that is trivial everywhere and has the group $\mathbb{Z}$ in degree $n$.

$$\begin{array}{rl}[B^n\mathbb{Z}]& =(\cdots \to [B^n\mathbb{Z}{]}_{n+1}\to [B^n\mathbb{Z}{]}_n\to [B^n\mathbb{Z}{]}_{n-1}\cdots \to [B^n\mathbb{Z}{]}_1\overrightarrow{\to }[B^n\mathbb{Z}{]}_0)\\ & =(\cdots \to *\to \mathbb{Z}\to *\cdots \to *\overrightarrow{\to }*)\end{array}.$$\begin{aligned} [\mathbf{B}^n \mathbb{Z}] &= ( \cdots \to [\mathbf{B}^n\mathbb{Z}]_{n+1} \to [\mathbf{B}^n\mathbb{Z}]_{n} \to [\mathbf{B}^n\mathbb{Z}]_{n-1} \cdots \to [\mathbf{B}^n\mathbb{Z}]_{1} \stackrel{\to}{\to} [\mathbf{B}^n\mathbb{Z}]_{0}) \\ &= ( \cdots \to {*} \to \mathbb{Z} \to {*} \cdots \to {*} \stackrel{\to}{\to} {*}) \end{aligned} \,.

Under the Quillen equivalence

$$\mid -\mid :\mathrm{\infty }\mathrm{Grpds}\to \mathrm{Top}$$|-| : \infty Grpds \to Top

between infinity-groupoids and topological spaces (see homotopy hypothesis) this sequence of suspensions of $\mathbb{Z}$ maps to the sequence of Eilenberg–Mac Lane spaces

$$\mid B^n\mathbb{Z}\mid \simeq K(\mathbb{Z},n)$$|\mathbf{B}^n \mathbb{Z}| \simeq K(\mathbb{Z}, n)

that give the Eilenberg–Mac Lane spectrum

$$(E_n):=(K(\mathbb{Z},n))$$(E_n) := (K(\mathbb{Z},n))

its name.

Related concepts

• K-theory spectrum

• cohomology theory