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\begin{document}

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\section*{Eilenberg-Mac Lane spectrum}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{incarnations}{Incarnations}\dotfill \pageref*{incarnations} \linebreak
\noindent\hyperlink{AsSymmetricSpectra}{As symmetric/orthogonal spectra}\dotfill \pageref*{AsSymmetricSpectra} \linebreak
\noindent\hyperlink{as_symmetric_monoidal_groupoids}{As symmetric monoidal $\infty$-groupoids}\dotfill \pageref*{as_symmetric_monoidal_groupoids} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{chromatic_filtration}{Chromatic filtration}\dotfill \pageref*{chromatic_filtration} \linebreak
\noindent\hyperlink{ordinary_homology_spectra_split}{Ordinary homology spectra split}\dotfill \pageref*{ordinary_homology_spectra_split} \linebreak
\noindent\hyperlink{fibrantcofibrant_models}{Fibrant-cofibrant models}\dotfill \pageref*{fibrantcofibrant_models} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $A$ an [[abelian group]], the \emph{Eilenberg-Mac Lane spectrum} $H A$ is the [[spectrum]] that [[Brown representability theorem|represents]] the [[ordinary cohomology|ordinary]] [[cohomology theory]]/[[ordinary homology]] 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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{incarnations}{}\subsection*{{Incarnations}}\label{incarnations}

\hypertarget{AsSymmetricSpectra}{}\subsubsection*{{As symmetric/orthogonal spectra}}\label{AsSymmetricSpectra}

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]].

\begin{defn}
\label{ReducedALinearizationOfnSphere}\hypertarget{ReducedALinearizationOfnSphere}{}
For $A$ an [[abelian group]] and $n \in \mathbb{N}$, the \textbf{reduced $A$-linearization} $A[S^n]_\ast$ of the [[n-sphere]] $S^n$ is the [[topological space]], whose underlying set is the [[quotient]]

\begin{displaymath}
\underset{k \in \mathbb{N}}{\sqcup}
   A^k \times (S^n)^k
  \longrightarrow
  A[S^n]_\ast
\end{displaymath}
of the [[tensor product]] with $A$ of the [[free abelian group]] on the underlying set of $S^n$, by the relation that identifies every [[formal linear combination]] of the (any fixed) basepoint of $S^n$ with 0. The [[topological space|topology]] is the induced [[quotient topology]] (of the [[disjoint union]] of [[product topological spaces]], where $A$ is equipped with the [[discrete topology]]).

\end{defn}
(\href{Eilenberg-MacLane+space#AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, def. 6.4.20})

\begin{prop}
\label{ReducedALinearizationOfnSphereIsEMSpace}\hypertarget{ReducedALinearizationOfnSphereIsEMSpace}{}
For $A$ a [[countable set|countable]] [[abelian group]], then the reduced $A$-linearization $A[S^n]_\ast$ (def. \ref{ReducedALinearizationOfnSphere}) is an [[Eilenberg-MacLane space]], in that its [[homotopy groups]] are

\begin{displaymath}
\pi_q(A[S^n]_\ast)
   \simeq
   \left\{
     \itexarray{
       A & if \; q = n
       \\
       \ast & otherwise
     }
   \right.
\end{displaymath}
(in particular for $n \geq 1$ then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).

\end{prop}
(\href{Eilenberg-MacLane+space#AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, corollary 6.4.23})

\begin{defn}
\label{}\hypertarget{}{}
For $A$ a [[countable set|countable]] [[abelian group]], then the [[orthogonal spectrum]] incarnation of the \textbf{[[Eilenberg-MacLane spectrum]]} $H A$ is the [[orthogonal spectrum]] with

\begin{itemize}%
\item components spaces

\begin{displaymath}
(H A)_V \coloneqq A[S^V]_\ast
\end{displaymath}
being the reduced $A$-linearization (def. \ref{ReducedALinearizationOfnSphere}) of the [[representation sphere]] $S^V$;

hence for $V = \mathbb{R}^n$ then

\begin{displaymath}
(H A)_n = A[S^n]_\ast
\end{displaymath}

\item $O(V)$-[[action]] on $A[S^V]_\ast$ induced from the canonical $O(V)$-action on $S^V$ ([[representation sphere]]);


\item structure maps

\begin{displaymath}
\sigma_{V,W}
    \;\colon\;
  (H A)_V \wedge S^W 
    \longrightarrow
  (H A)_{V\oplus W}
\end{displaymath}
hence

\begin{displaymath}
A[S^V] \wedge S^W
    \longrightarrow
  A[S^{V \oplus W}]
\end{displaymath}
given by

\begin{displaymath}
\left(
  \left(
     \underset{i}{\sum} a_i x_i
  \right),
  y
  \right)
    \mapsto
  \underset{i}{\sum} a_i (x_i, y)
  \,.
\end{displaymath}


\end{itemize}
The incarnation of $H A$ as a [[symmetric spectrum]] is the same, with the group action of $O(n)$ replaced by the [[subgroup]] action of the [[symmetric group]] $\Sigma(n) \hookrightarrow O(n)$.

If $R$ is a [[commutative ring]], then the Eilenberg-MacLane spectrum $H R$ becomes a commutative [[orthogonal ring spectrum]] (or [[symmetric ring spectrum]], respectively) by

\begin{enumerate}%
\item taking the multiplication

\begin{displaymath}
(H R)_{V_1} \wedge (H R)_{V_2}
    =
  R[S^{V_1}]_\ast \wedge R[S^{V_2}]_\ast 
    \longrightarrow
  R[S^{V_1 \oplus V_2}]
   =
  (H R)_{V_1 \oplus V_2}
\end{displaymath}
to be given by

\begin{displaymath}
\left(
  \left(
     \underset{i}{\sum} a_i x_i
  \right)
  ,
  \left(
    \underset{j}{\sum} b_j y_j
  \right)
  \right)
   \;\mapsto\;
  \underset{i,j}{\sum}
   (a_i \cdot b_j)(x_i, y_j)
\end{displaymath}

\item taking the unit maps

\begin{displaymath}
S^V 
    \longrightarrow
  A[S^V]_\ast = (H R)_V
\end{displaymath}
to be given by the canonical inclusion of generators

\begin{displaymath}
x \mapsto 1 x
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
(\hyperlink{Schwede12}{Schwede 12, example I.1.14}, \hyperlink{Schwede15}{Schwede 15, V, costruction 3.21})

\hypertarget{as_symmetric_monoidal_groupoids}{}\subsubsection*{{As symmetric monoidal $\infty$-groupoids}}\label{as_symmetric_monoidal_groupoids}

Under the identification of [[connective spectra]] with ``[[abelian infinity-groups]]'' the Eilenberg-MacLane spectrum $H A$ simply \emph{is} the group $A$.

Here the set $A$ is regarded as a [[discrete category|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]]

\begin{displaymath}
A, \mathbf{B} A, \mathbf{B}^2 A, \mathbf{B}^3 A, \cdots
\end{displaymath}
that in this case happen to be [[strict omega-groupoids]]. The [[strict omega-groupoid]] $\mathbf{B}^n A$ has only [[identity]] $k$-morphisms for all $k$, except for $k = n$, where $\mathrm{Mor}_n(\mathbf{B}^n A) = A$ are the endomorphisms of the unique identity $(n-1)$-morphism.

The [[strict ∞-groupoid]] $\mathbf{B}^n A$ 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{displaymath}
\begin{aligned}
    [\mathbf{B}^n A] &=
     (  \cdots \to [\mathbf{B}^n A]_{n+1} \to [\mathbf{B}^n A]_{n} \to 
     [\mathbf{B}^n A]_{n-1} \cdots \to [\mathbf{B}^n A]_{1} \stackrel{\to}{\to} [\mathbf{B}^n A]_{0})
      \\ &=
    (  \cdots \to {*} \to A \to 
    {*} \cdots \to {*} \stackrel{\to}{\to} {*})
  \end{aligned}
  \,.
\end{displaymath}
Under the [[Quillen equivalence]]

\begin{displaymath}
|-| : \infty Grpds \to Top
\end{displaymath}
between [[infinity-groupoids]] and [[topological spaces]] (see \emph{[[homotopy hypothesis]]}) this sequence of suspensions of $A$ maps to the sequence of [[Eilenberg?Mac Lane spaces]]

\begin{displaymath}
|\mathbf{B}^n A|
  \simeq
  K(A, n)
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{chromatic_filtration}{}\subsubsection*{{Chromatic filtration}}\label{chromatic_filtration}

[[!include chromatic tower examples - table]]

\hypertarget{ordinary_homology_spectra_split}{}\subsubsection*{{Ordinary homology spectra split}}\label{ordinary_homology_spectra_split}

\textbf{[[ordinary homology spectra split]]}: For $S$ any [[spectrum]] and $H A$ an Eilenberg-MacLane spectrum, then the [[smash product]] $S\wedge H A$ (the $A$-[[ordinary homology]] spectrum) is non-canonically equivalent to a product of EM-spectra (hence a [[wedge sum]] of EM-spectra in the finite case).

\hypertarget{fibrantcofibrant_models}{}\subsubsection*{{Fibrant-cofibrant models}}\label{fibrantcofibrant_models}

\href{http://mathoverflow.net/a/218069/381}{MO comment}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[generalized Eilenberg-MacLane spectrum]]


\item [[Moore spectrum]]


\item [[cohomology theory]], [[ordinary cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Textbook accounts include

\begin{itemize}%
\item [[Frank Adams]], part III, section 2 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Stanley Kochmann]], section 3.5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Peter May]], chapter 22 of \emph{A concise course in algebraic topology} (\href{http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf}{pdf})


\item [[John Rognes]], section 3.2 3.4 of \emph{The Adams spectral sequence}, 2012 (\href{http://folk.uio.no/rognes/papers/notes.050612.pdf}{pdf})



\end{itemize}
As [[symmetric spectra]]

\begin{itemize}%
\item [[Stefan Schwede]], Example I.1.14 in \emph{Symmetric spectra}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec.pdf}{pdf})


\item [[John Greenlees]], example 3.6, example 4.16 in \emph{Spectra for commutative algebraists} (\href{http://www.math.uic.edu/~bshipley/greenlees.SpectraMSRI.pdf}{pdf})



\end{itemize}
as [[orthogonal spectra]]:

\begin{itemize}%
\item [[Stefan Schwede]], construction V 3.21 in \emph{[[Global homotopy theory]]}, 2015

\end{itemize}
[[!redirects Eilenberg-MacLane spectrum]] [[!redirects Eilenberg--MacLane spectrum]] [[!redirects Eilenberg--Mac Lane spectrum]] [[!redirects Eilenberg?MacLane spectrum]] [[!redirects Eilenberg?Mac Lane spectrum]] [[!redirects Eilenberg-MacLane spectra]] [[!redirects Eilenberg-Mac Lane spectra]] [[!redirects Eilenberg--MacLane spectra]] [[!redirects Eilenberg--Mac Lane spectra]] [[!redirects Eilenberg?MacLane spectra]] [[!redirects Eilenberg?Mac Lane spectra]]

[[!redirects HR]]

[[!redirects HA]] [[!redirects HC]]



\end{document}