nLab
Eilenberg-Mac Lane spectrum

As a symmetric monoidal infinity-groupoid the Eilenberg–Mac Lane spectrum is the integers

$$E_{\mathrm{EM}}:=\mathbb{Z}.$$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 additve group structure on the integers.

Accordingly, the infinite tower of suspensions induced by this is the sequence of infinity-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 $\mathrm{\infty }$-grouopoid $B^n\mathbb{Z}$ 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 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.