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\begin{document}

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\section*{Eilenberg-Mac Lane object}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_}{In $Top$}\dotfill \pageref*{in_} \linebreak
\noindent\hyperlink{in_sheaf_toposes}{In $(\infty,1)$-sheaf $(\infty,1)$-toposes}\dotfill \pageref*{in_sheaf_toposes} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The notion of \emph{Eilenberg--Mac Lane object} in an [[(∞,1)-topos]] or [[stable (∞,1)-category]] generalizes the notion of [[Eilenberg?Mac Lane space]] from the [[(∞,1)-topos]] [[Top]] of [[topological space]]s or the [[stable (∞,1)-category]] of [[spectrum|spectra]]:

it is an object $\mathbf{B}^n A$ obtained from an abelian [[group object]] $A$ by [[delooping]] that $n$ times.

An object that is both $n$-[[truncated]] as well as $n$-connected.

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

\begin{udefn}
Let $\mathbf{H}$ be an [[(∞,1)-topos]].

For $n \in \mathbb{N}$ an \textbf{Eilenberg-MacLane object} $X$ of degree $n$

\begin{itemize}%
\item a [[pointed object]] $* \to X \in \mathbf{H}$


\item which is both $n$-[[n-connective|connective]] as well as $n$-[[truncated]].



\end{itemize}
\end{udefn}
This appears as [[Higher Topos Theory|HTT, def. 7.2.2.1]]

\begin{uremark}
If one drops the condition that $X$ has a global point, then this is the definition of [[∞-gerbe]]s.

\end{uremark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The next proposition asserts that Eilenberg-MacLane objects defined this way are shifted [[group object in an (∞,1)-category|(∞,1)-categorical group objects]]:

\begin{uprop}
For $\mathbf{H}$ an [[(∞,1)-topos]], $\mathbf{H}_*$ its [[(∞,1)-category]] of [[pointed object]]s, $Disc(\mathbf{H})$ the full sub-[[(∞,1)-category]] on discrete objects (0-[[truncated]] objects) and $n \in \mathbb{N}$, write

\begin{displaymath}
\pi_n : \mathbf{H}_* \to Disc(\mathbf{H}_*)
\end{displaymath}
for the [[(∞,1)-functor]] that assigns the $n$-th [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]].

\begin{itemize}%
\item For $n = 0$ this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and [[pointed object]]s of $Disc(\mathbf{H})$; moreover, the restriction $\pi_0:Disc(\mathbf{H}_*)\to Disc(\mathbf{H}_*)$ is equivalent to the identity.


\item For $n = 1$ this establishes an equivalence between the full subcategory on degree 1 Eilenberg-MacLane objects and the category of [[group object in an (∞,1)-category|group objects]] in $Disc(\mathbf{H})$.


\item For $n \geq 2$ this establishes an equivalence between the full subcategory on degree $n$ Eilenberg-MacLane objects and the category of commutative [[group object in an (∞,1)-category|group objects]] in $Disc(\mathbf{H})$.



\end{itemize}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop. 7.2.2.12]].

\end{proof}
\begin{udef}
For $\mathbf{H}$ an [[(∞,1)-topos]] and $n \in \mathbb{N}$ write $K(-,n)$ for [[generalized the|the]] homotopy inverse to the equivalence induced by $\pi_n$ by the above proposition. For $A \in Disc(\mathbf{H})$ an (abelian) group object we say that

\begin{displaymath}
K(A,n) \in \mathbf{H}
\end{displaymath}
is the degree $n$-Eilenberg-MacLane object of $A$.

\end{udef}
\begin{uprop}
We have that

\begin{displaymath}
K(A,n) \simeq \mathbf{B}^n A
\end{displaymath}
is [[generalized the|the]] $n$-fold [[delooping]] of the discrete group object $A$.

\end{uprop}
\begin{proof}
\begin{quote}%
check


\end{quote}
The [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] are defined in terms of the canonical [[power]]ing of $\mathbf{H}$ over [[∞Grpd]]

\begin{displaymath}
(-)^{(-)} : \infty Grpd \times \mathbf{H} \to \mathbf{H}
  \,.
\end{displaymath}
For fixed [[∞-groupoid]] $K$ this

\begin{displaymath}
(-)^{K} : \mathbf{H} \to \mathbf{H}
  \,.
\end{displaymath}
preserves $(\infty,1)$-[[limit]]s and hence [[pullback]]s. It follows that the categorical homotopy groups of the [[loop space object]] $\Omega K(A,n)$ are those of $K(A,n)$, shifted down by one degree.

By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies $\Omega K(A,n) \simeq K(A,n-1)$.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_}{}\subsubsection*{{In $Top$}}\label{in_}

In the archetypical [[(∞,1)-topos]] [[Top]]$\simeq$ [[∞Grpd]] the notion of Eilenberg-MacLane object reduces to the traditional notion of [[Eilenberg-MacLane space]].

\hypertarget{in_sheaf_toposes}{}\subsubsection*{{In $(\infty,1)$-sheaf $(\infty,1)$-toposes}}\label{in_sheaf_toposes}

Recall that an [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]]/[[∞-stack]] [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$ may be [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial sheaves]] on $C$.

In terms of this model the Eilenberg-Mac Lane objects $K(A,n) \in \mathbf{H}$ (for abelian $A$) are the \textbf{Eilenberg-MacLane [[sheaf|sheaves]]} of [[abelian sheaf cohomology]] theory.

Under the [[Dold?Kan correspondence]]

\begin{displaymath}
N : sAb \stackrel{\leftarrow}{\to} Ch_+ : \Gamma
\end{displaymath}
[[chain complex]]es $A[n]$ of abelian groups concentrated in degree $n$ map into [[simplicial set]]s

\begin{displaymath}
K(A,n) := \Gamma(A[n])
\end{displaymath}
and these to the corresponding constant simplicial sheaves on the [[site]] $C$, that we denote by the same symbol, for convenience.

Under the equivalence

\begin{displaymath}
\mathbf{H} = Sh_{(\infty,1)}(C) \simeq (sSh(C)_{loc})^\circ
\end{displaymath}
of $\mathbf{H}$ with the [[Kan complex]]-enriched full subcategory of $sSh(C)$ on fibrant cofibrant objects, this identifies the fibrant reeplacement -- the [[∞-stackification]] -- of $\Gamma(A[n])$ with the Eilenberg-MacLane object in $\mathbf{H}$.

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

The notion of [[cohomology]] in the [[(∞,1)-topos]] $\mathbf{H}$ with coefficients in an object $\mathcal{A} \in \mathbf{H}$ is often taken to be restricted to the case where $\mathcal{A}$ is an Eilenberg-MacLane object.

For $A \in Disc(\mathbf{A})$ an abelian group object, and $n \in \mathbb{N}$, the degree $n$-cohomology of an object $X \in \mathbf{H}$ is the cohomology with coefficients in $K(A,n)$:

\begin{displaymath}
H^n(X, A) := H(X, K(A,n)) := \pi_0 \mathbf{H}(X, K(A,n))
  \,.
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The general discussion of Eilenberg-MacLane objects is in section 7.2.2 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
For a discussion of Eilenberg-MacLane objects in the context of the [[model structure on simplicial presheaves]] see top of page 4 of

\begin{itemize}%
\item [[Rick Jardine]], \emph{Fields Lectures: Simplicial Presheaves} (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{pdf})

\end{itemize}
Discussion in [[equivariant homotopy theory]] (see also at \emph{[[Bredon cohomology]]}) is in

\begin{itemize}%
\item \{Lewis92\} L. G. Lewis, \emph{Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems}, Topology Appl., 48 (1992), no. 1, pp. 25--61.

\end{itemize}
Formalization in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Dan Licata]], [[Eric Finster]], \emph{Eilenberg-MacLane spaces in homotopy type theory}, LICS 2014 (\href{http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf}{pdf text}, \href{https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda}{Agda HoTT code}, \href{http://homotopytypetheory.org/2014/04/15/eilenberg-maclane-spaces-in-hott/}{web discussion})

\end{itemize}
[[!redirects Eilenberg-MacLane object]] [[!redirects Eilenberg?MacLane object]] [[!redirects Eilenberg--MacLane object]] [[!redirects Eilenberg-Mac Lane object]] [[!redirects Eilenberg?Mac Lane object]] [[!redirects Eilenberg--Mac Lane object]] [[!redirects Eilenberg-MacLane objects]] [[!redirects Eilenberg?MacLane objects]] [[!redirects Eilenberg--MacLane objects]] [[!redirects Eilenberg-Mac Lane objects]] [[!redirects Eilenberg?Mac Lane objects]] [[!redirects Eilenberg--Mac Lane objects]]



\end{document}