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\begin{document}

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\section*{Bousfield-Kan map}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category 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{bousfieldkan_for_simplicial_objects}{Bousfield--Kan for simplicial objects}\dotfill \pageref*{bousfieldkan_for_simplicial_objects} \linebreak
\noindent\hyperlink{bousfieldkan_for_cosimplicial_objects}{Bousfield--Kan for cosimplicial objects}\dotfill \pageref*{bousfieldkan_for_cosimplicial_objects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_homotopy_limits}{Relation to homotopy limits}\dotfill \pageref*{relation_to_homotopy_limits} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The Bousfield--Kan map(s) are comparison morphisms in a [[simplicial model category]] between two different ``puffed up'' versions of (co)limits over (co-)simplicial objects: one close to a [[homotopy limit|homotopy (co)limit]] and the other a version of nerve/geometric realization.

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

Let $C$ be an ([[SimpSet|SSet]],$\otimes = \times$)-[[enriched category]] and write

\begin{displaymath}
\Delta : \Delta \to Set^{\Delta^{op}}
\end{displaymath}
for the canonical cosimplicial [[simplicial set]] (the [[adjunct]] of the [[hom-functor]] $\Delta^{op} \times \Delta \to Set$).

Write furthermore $N(\Delta/-) : \Delta \to Set^{\Delta^{op}}$ for the [[fat simplex]], the [[cosimplicial object|cosimplicial]] [[simplicial set]] which assigns to $[n]$ the [[nerve]] of the [[overcategory]] $\Delta / [n]$.

The \textbf{Bousfield--Kan map of cosimplicial simplicial maps} is a canonical morphism

\begin{displaymath}
\varphi : N(\Delta/(-)) \to \Delta
\end{displaymath}
of cosimplicial simplicial sets.

This can also be regarded as a morphism

\begin{displaymath}
\varphi : N((-)/\Delta^{op})^{op} \to \Delta
  \,.
\end{displaymath}
This morphism induces the following morphisms between (co)[[simplicial object]]s in $C$.

\hypertarget{bousfieldkan_for_simplicial_objects}{}\subsubsection*{{Bousfield--Kan for simplicial objects}}\label{bousfieldkan_for_simplicial_objects}

For $X : \Delta^\op \to C$ any [[simplicial object]] in $C$, the \textbf{realization} of $X$ is the [[end|coend]]

\begin{displaymath}
|X| := X \otimes_{\Delta^{op}} \Delta := 
  \int^{[n] \in \Delta} X_n \otimes \Delta^n
  \,,
\end{displaymath}
where in the integrand we have the [[copower]] or tensor of $C$ by [[SimpSet|SSet]].

Here the \textbf{Bousfield--Kan} map is the morphism

\begin{displaymath}
X \otimes_{\Delta^{op}} N((-)/\Delta^{op})^{op}
  \stackrel{Id_X \otimes_{\Delta^{op}} \phi }{\to}
  X \otimes_{\Delta^{op}} \Delta
  \,.
\end{displaymath}
\hypertarget{bousfieldkan_for_cosimplicial_objects}{}\subsubsection*{{Bousfield--Kan for cosimplicial objects}}\label{bousfieldkan_for_cosimplicial_objects}

For $X : \Delta \to C$ any cosimplicial object, its \textbf{totalization} is the $\Delta$-[[weighted limit]]

\begin{displaymath}
Tot X := lim^\Delta X \simeq \int_{[n \in \Delta]} X_n^{\Delta^n}
  \,,
\end{displaymath}
where in the integrand we have the [[power]] or cotensor $X_n^{\Delta^n} = \pitchfork(\Delta, X_n)$ of $C$ by [[SimpSet|SSet]].

Here the \textbf{Bousfield--Kan} morphism is the morphism

\begin{displaymath}
Tot X \simeq hom(\Delta,X) \stackrel{hom(\phi,Id_X)}{\to}  
  hom(N(\Delta/(-)), X)
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{utheorem}
If the [[simplicial object]] $X$ is [[Reedy category|Reedy cofibrant]] then its Bousfield--Kan map is a natural weak equivalence.

If the co[[simplicial object]] $X$ is [[Reedy category|Reedy fibrant]] then its Bousfield--Kan map is a natural weak equivalence.

\end{utheorem}
\begin{proof}
This can be proven for instance using [[homotopy colimit]]s in the [[Reedy model structure]]. Details are at .

\end{proof}
\hypertarget{relation_to_homotopy_limits}{}\subsection*{{Relation to homotopy limits}}\label{relation_to_homotopy_limits}

When the co[[simplicial object]] $X$ is degreewise fibrant, then

\begin{displaymath}
lim^{N(\Delta/(-))} X
  \simeq holim X
\end{displaymath}
computes the [[homotopy limit]] of $X$ as a [[weighted limit]] (as explained there). Then the above theorem says that the homotopy limit is already computed by the totalization

\begin{displaymath}
holim X \simeq lim^\Delta X
  \,.
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original reference is

\begin{itemize}%
\item [[Aldridge Bousfield]] and [[Dan Kan]], \emph{Homotopy limits, completions and localizations} Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

\end{itemize}
Reviews include

\begin{itemize}%
\item Hirschhorn, \emph{Simplicial model categories and their localization}.

\end{itemize}
The Bousfield--Kan map(s) are on p. 397, def. 18.7.1 and def. 18.7.3.

\emph{Realization} and \emph{totalization} are defs 18.6.2 and 18.6.3 on p. 395.

Notice that this book writes $B$ for the nerve!



\end{document}