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\begin{document}

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\section*{totalization}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - 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{as_an_end}{As an end}\dotfill \pageref*{as_an_end} \linebreak
\noindent\hyperlink{AsTheHomotopyLimit}{As the homotopy limit}\dotfill \pageref*{AsTheHomotopyLimit} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_and_homology}{Homotopy and homology}\dotfill \pageref*{homotopy_and_homology} \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}

The \emph{totalization} of a [[cosimplicial object]] is the [[duality|dual]] concept to the [[geometric realization]] of a [[simplicial object]].

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

\hypertarget{as_an_end}{}\subsubsection*{{As an end}}\label{as_an_end}

For $A : \Delta \to C$ a [[cosimplicial object]] in a [[category]] $C$ which is [[powering|powered]] over [[simplicial sets]] and for

\begin{displaymath}
\Delta : [n] \mapsto \Delta[n]
\end{displaymath}
the canonical cosimplicial simplicial set of [[simplices]], the \textbf{totalization} of $A$ is the [[end]]

\begin{equation}
\int_{[k]\in \Delta}
  (A_k)^{\Delta[k]}
  \,\,\,
  \in 
  C
  \,.
\label{AsAnEnd}\end{equation}
\hypertarget{AsTheHomotopyLimit}{}\subsubsection*{{As the homotopy limit}}\label{AsTheHomotopyLimit}

For a [[cosimplicial object]] $A \colon \Delta \to \mathcal{C}$ in a suitable [[model category]] such that $A$ is a [[fibrant object]] with respect to the [[Reedy model structure]] on $Func(\Delta, \mathcal{C})$, totalization in terms of the [[end]]-construction above in \eqref{AsAnEnd} is a model for the [[homotopy limit]] over $A$.

(\hyperlink{Hirschhorn15}{Hirschhorn 15, theorem 9.2})

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

\hypertarget{homotopy_and_homology}{}\subsubsection*{{Homotopy and homology}}\label{homotopy_and_homology}

The [[homotopy groups]] of the totalization of a [[cosimplicial space]] are computed by a [[Bousfield-Kan spectral sequence]].

The [[homology groups]] by an [[Eilenberg-Moore spectral sequence]].

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

Formally the dual to totalization is [[geometric realization]]: where totalization is the [[end]] over a [[power]]ing with $\Delta$, realization is the [[coend]] over the [[tensoring]].

But various other operations carry names similar to ``totalization''. For instance a [[total chain complex]] is related under [[Dold-Kan correspondence]] to the [[diagonal]] of a [[bisimplicial set]] -- see [[Eilenberg-Zilber theorem]]. As discussed at \emph{[[bisimplicial set]]}, this is [[weak homotopy equivalence|weakly homotopy equivalent]] to the operation that is often called $Tot$ and called the \emph{total simplicial set} of a bisimplicial set.

To a cosimplicial chain complex we can assign a double complex by taking the alternating sum of the coface maps. Then the totalization of this cosimplicial object and the totalization of the double complex as defined in homological algebra coincide. Moreover, the associated [[Bousfield-Kan spectral sequence]] and [[spectral sequence of a double complex]] coincide.

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

The concept [[cosimplicial spaces]] originates with

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Daniel Kan]], chapter X.3 of \emph{[[Homotopy limits, completions and localizations]]}, Lecture Notes in Mathematics, Vol 304, Springer 1972

\end{itemize}
Quick review includes

\begin{itemize}%
\item [[Daniel Dugger]], section 5.3 of \emph{A primer on homotopy colimits} (\href{https://pages.uoregon.edu/ddugger/hocolim.pdf}{pdf})

\end{itemize}
The generalization to [[cosimplicial objects]] in more general [[model categories]] is discussed in

\begin{itemize}%
\item [[Aldridge Bousfield]], \emph{Cosimplicial resolutions and homotopy spectral sequences in model categories}, Geom. Topol. 7 (2003) 1001-1053 (\href{http://arxiv.org/abs/math/0312531}{arXiv:math/0312531})

\end{itemize}
Review of this includes

\begin{itemize}%
\item [[Marc Levine]], \emph{The Adams-Novikov spectral sequence and Voevodsky's slice tower}, Geom. Topol. 19 (2015) 2691-2740 (\href{http://arxiv.org/abs/1311.4179}{arXiv:1311.4179})

\end{itemize}
Some kind of notes are in

\begin{itemize}%
\item Rosona Eldred, \emph{Tot primer} (\href{https://drive.google.com/file/d/0B6WoYElpsU2TTXdNbmNyXzZjamc/view}{pdf})

\end{itemize}
Discussion of totalizations as [[homotopy limits]] includes

\begin{itemize}%
\item [[Philip Hirschhorn]], section 9 of \emph{The diagonal of a multicosimplicial object} (\href{https://arxiv.org/abs/1506.06837}{arXiv:1506.06837})


\item [[Akhil Mathew]], [[Vesna Stojanoska]], \emph{Fibers of partial totalizations of a pointed cosimplicial space}, Proc. Amer. Math. Soc. 144 (2016), no. 1, 445--458 (\href{https://arxiv.org/abs/1408.1665}{arXiv:1408.1665})



\end{itemize}
[[!redirects totalizations]]



\end{document}