Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 2.5.6.9 (Eilenberg-MacLane Spaces). Let $G$ be an abelian group, let $n$ be a nonnegative integer, and let $G[n]$ denote the chain complex consisting of the single abelian group $G$, concentrated in degree $n$ (Example 2.5.1.2). We will denote the simplicial abelian group $\mathrm{K}( G[n] )$ by $\mathrm{K}(G,n)$ and refer to it as the $n$th Eilenberg-MacLane space of $G$.

For small values of $n$, it will be useful to consider allow more general coefficients.

  • If $G$ is any group (not necessarily abelian), we let $\mathrm{K}(G,1)$ denote the classifying simplicial set $B_{\bullet }(G)$ (Construction 1.3.2.5).

  • If $G$ is any set, we let $\mathrm{K}(G,0)$ denote the constant simplicial set $\underline{G}$ (Construction 1.1.5.2).

By virtue of Examples 2.5.6.7 and 2.5.6.8, this recovers the first definition in the case where $G$ is an abelian group.