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\begin{document}

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\section*{bisimplicial set}

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

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

[[!include homotopy - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Diagonal}{Diagonal}\dotfill \pageref*{Diagonal} \linebreak
\noindent\hyperlink{TotalSimplicialSets}{Total d\'e{}calage and total simplicial sets}\dotfill \pageref*{TotalSimplicialSets} \linebreak
\noindent\hyperlink{geometric_realization}{Geometric realization}\dotfill \pageref*{geometric_realization} \linebreak
\noindent\hyperlink{model_structures}{Model structures}\dotfill \pageref*{model_structures} \linebreak
\noindent\hyperlink{induced_from_the_diagonal}{Induced from the diagonal}\dotfill \pageref*{induced_from_the_diagonal} \linebreak
\noindent\hyperlink{induced_from_codiagonal_}{Induced from codiagonal $\nabla$.}\dotfill \pageref*{induced_from_codiagonal_} \linebreak
\noindent\hyperlink{remark_on_notation}{Remark on notation}\dotfill \pageref*{remark_on_notation} \linebreak
\noindent\hyperlink{bisimplicial_abelian_groups}{Bisimplicial abelian groups}\dotfill \pageref*{bisimplicial_abelian_groups} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A bisimplicial set is a [[bisimplicial object]] in [[Set]].

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

\hypertarget{Diagonal}{}\subsubsection*{{Diagonal}}\label{Diagonal}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(diagonal)}

For $X_{\bullet,\bullet}$ a bisimplicial set, its \textbf{diagonal} is the simplicial set that is the precomposition with $(Id, Id) : \Delta^{op} \to \Delta^{op} \times \Delta^{op}$, i.e. the simplicial set with components

\begin{displaymath}
d(X)_n = X_{n,n}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(realization)}

The \textbf{realization} $|X|$ of a bisimplicial set $X_{\bullet,\bullet}$ is the [[simplicial set]] that is given by the [[coend]]

\begin{displaymath}
|X| =  \int^{[n] \in \Delta} X_{n, *} \times \Delta[n]
\end{displaymath}
in [[sSet]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(diagonal is realization)}

For $X$ a bisimplicial set, its diagonal $d(X)$ is (isomorphic to) its realization $|X|$:

\begin{displaymath}
|X| \simeq d(X)
  \,.
\end{displaymath}
\end{prop}
This is for instance exercise 1.6 in in \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi}{chapter 4} \hyperlink{GoerssJardine}{Goerss-Jardine}. For a derivation see the examples at \emph{[[homotopy colimit]]}.

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(diagonal is homotopy colimit)}

The diagonal of a bisimplicial set $X_{\bullet,\bullet}$ is also (up to weak equivalence) the [[homotopy colimit]] of $X$ regarded as a simpliciall diagram in the [[model structure on simplicial sets]]

\begin{displaymath}
diag X \simeq hocolim (X : \Delta^{op} \to sSet_{Quillen})
  \,.
\end{displaymath}
\end{prop}
This appears for instance as theorem 3.6 in (\hyperlink{Isaacson}{Isaacson}).

\begin{proof}
This follows with the above equivalence to the [[coend]] $diag X \simeq \int^{[k] \in \Delta} \Delta[k] \cdot X_k$ and general expression of [[homotopy colimit]]s by coends (as discussed there) in terms of the [[Quillen bifunctor]]

\begin{displaymath}
\int^\Delta (-) \cdot (-)
    :
   [\Delta, sSet_{Quillen}]_{Reedy} \times
   [\Delta, sSet_{Quillen}]_{Reedy}  
  \to 
    sSet_{Quillen}
\end{displaymath}
in [[Reedy model structure]]s (as discussed there) by using that $\Delta[-] : \Delta \to sSet_{Quillen}$ is a Reedy cofibrant resultion of the point in $[\Delta, sSet_{Quillen}]$ and that every object in $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$ is cofibrant.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(degreewise weak equivalences)}

Let $X,Y : \Delta^{op} \times \Delta^{op} \to Set$ be bisimplicial sets. A morphism $f : X \to Y$ which is degreewise in one argument a weak equivalence $f_{n,\bullet} : X(n,\bullet) \to Y(n,\bullet)$ induces a weak equivalence $d(f) : d(X) \to d(Y)$ of the associated diagonal simplicial sets (with respect to the standard [[model structure on simplicial sets]]).

\end{prop}
\begin{proof}
This is prop 1.9 in \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi}{chapter 4} of

\begin{itemize}%
\item Goerss-Jardine, \emph{Simplicial Homotopy Theory} (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{dvi})

\end{itemize}
\end{proof}
\hypertarget{TotalSimplicialSets}{}\subsubsection*{{Total d\'e{}calage and total simplicial sets}}\label{TotalSimplicialSets}

There is a [[functor]] called \emph{[[ordinal sum]]} (see also at [[simplex category]])

\begin{displaymath}
+ : \Delta^\op \times \Delta^{op} \to \Delta^{op}
  \,.
\end{displaymath}
\begin{displaymath}
+ : [k], [l] \mapsto [k+l+1]
  \,.
\end{displaymath}
This induces an [[adjoint triple]]

\begin{displaymath}
ssSet
   \stackrel{\overset{+_!}{\longrightarrow}}{\stackrel{\overset{+^*}{\longleftarrow}}{\underset{+_*}{\longrightarrow}}}
  sSet
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
Here

\begin{itemize}%
\item $T \coloneqq +_*$ is called the \textbf{total simplicial set} functor or \textbf{Artin-Mazur codiagonal} (we will use the first of these as codiagonal also has another accepted meaning, see [[codiagonal]]);


\item $Dec \coloneqq +^*$ is called the [[total décalage]] functor (inside which is plain [[décalage]]);



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
$T$ preserves degreewise [[weak equivalences]] of [[simplicial sets]].

\end{prop}
\begin{prop}
\label{TotalAndDiagonal}\hypertarget{TotalAndDiagonal}{}
For $X$ any bisimplicial set

\begin{itemize}%
\item the canonical morphism

\begin{displaymath}
diag X \to T X
\end{displaymath}
from the [[diagonal]] to the total simplicial set is a [[weak equivalence]] in the [[model structure on simplicial sets]];


\item the [[unit of an adjunction|adjunction unit]]

\begin{displaymath}
X \to T Dec X
\end{displaymath}
is a weak equivalence.



\end{itemize}
\end{prop}
These statements are for instance in (\hyperlink{CegarraRemedios}{CegarraRemedios}) and (\hyperlink{Stevenson}{Stevenson}). They may be considered as a non-additive versions of the [[Eilenberg-Zilber theorem]].

\begin{remark}
\label{}\hypertarget{}{}
By prop. \ref{TotalAndDiagonal} and the usual \emph{[[Eilenberg-Zilber theorem]]} it follows that under forming chain complexes for [[simplicial homology]], total simplicial sets correspond to [[total complexes]] of [[double complexes]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
After [[geometric realization]] these spaces are even related by a [[homeomorphism]].

\end{remark}
(This seems to be due to Berger and H\"u{}bschmann, but related results were known to Zisman as they are so mentioned by Cordier in his work on homotopy limits.)

\begin{remark}
\label{}\hypertarget{}{}
The standard [[delooping]] functor for [[simplicial groups]]

\begin{displaymath}
\bar W : sGrp \to sSet_*
\end{displaymath}
is the composite

\begin{displaymath}
\bar W 
   : 
  sGrp
   \stackrel{\mathbf{B}}{\longrightarrow}
  sGrpd
   \stackrel{N}{\longrightarrow}
  ssSet
   \stackrel{T}{\longrightarrow}
  sSet
  \,.
\end{displaymath}
\end{remark}
We have the following explicit formula for $T X$, attributed to [[John Duskin]]:

\begin{lemma}
\label{}\hypertarget{}{}
For $X$ a [[bisimplicial set]] the total simplicial set $T X$ is in degree $n$ the [[equalizer]]

\begin{displaymath}
(T X)_n 
  \to
  \prod_{i = 0}^n X_{i, n-i}
  \stackrel{\longrightarrow}{\longrightarrow}
  \prod_{i = 0}^{n-1}
  X_{i, n-i-1}
\end{displaymath}
where the components of the two morphisms on the right are

\begin{displaymath}
\prod_{i = 0}^n X_{i,n-i}
   \stackrel{p_i}{\to}
  X_{i, n-i}
 \stackrel{d_0^v}{\to}
  X_{i, n-i-1}
\end{displaymath}
and

\begin{displaymath}
\prod_{i = 0}^n X_{i,n-i}
  \stackrel{p_{i+1}}{\to}
  X_{i+1,n-i-1}
  \stackrel{d_{i+1}^h}{\to}
  X_{i,n-i-1}
 \,.
\end{displaymath}
The face maps $d_i : (T X)_n \to (T X)_{n-1}$ are given by

\begin{displaymath}
d_i 
    =
  (d_i^v p_0, 
   d_{i-1}^v p_1,
   \cdots,
   d_1^v p_{i-1},
   d_i^h p_{i+1},
   d_i^h p_{i+2},
   \cdots,
   d_i^h p_n
  )
\end{displaymath}
and the degeneracy maps are given by

\begin{displaymath}
s_i 
  =
  (s_i^v p_0, s_{i-1}^v p_1, \cdots, 
   s_0^v p_i, s_i^h p_{i+1}, \cdots, 
   s_i^h p_n)
  \,.
\end{displaymath}
The $(Dec \dashv T)$-[[unit of an adjunction|adjunction unit]] $\eta : X \to T Dec X$ is given in degree $n$ by

\begin{displaymath}
\eta : x \mapsto 
   (s_0(x), s_1(x), \cdots, s_n(x))
  \,.
\end{displaymath}
\end{lemma}
\hypertarget{geometric_realization}{}\subsubsection*{{Geometric realization}}\label{geometric_realization}

See [[geometric realization of simplicial topological spaces]].

\hypertarget{model_structures}{}\subsection*{{Model structures}}\label{model_structures}

There are various useful [[model category]] structures on the category of bisimplicial sets.

\hypertarget{induced_from_the_diagonal}{}\subsubsection*{{Induced from the diagonal}}\label{induced_from_the_diagonal}

There is an [[adjunction]]

\begin{displaymath}
(L \dashv diag)   : 
  ssSet
  \stackrel{\overset{L}{\longleftarrow}}{\underset{diag}{\longrightarrow}}
  sSet 
  \,.
\end{displaymath}
The [[transferred model structure]] along this adjunction of the standard [[model structure on simplicial sets]] exists and with respect to it the above [[Quillen adjunction]] is a [[Quillen equivalence]].

This is due to (\hyperlink{Moerdijk}{Moerdijk 89})

\hypertarget{induced_from_codiagonal_}{}\subsubsection*{{Induced from codiagonal $\nabla$.}}\label{induced_from_codiagonal_}

The [[transferred model structure]] on $ssSet$ along the total simplicial set functor $T$ exists. And for it

\begin{displaymath}
(Dec \dashv T) : ssSet 
    \stackrel{\overset{Dec}{\longleftarrow}}{\underset{T}{\longrightarrow}}
   sSet
\end{displaymath}
is a [[Quillen equivalence]].

\begin{prop}
\label{}\hypertarget{}{}
Every diag-fibration is also a $T$-fibration.

\end{prop}
This is (\hyperlink{CegarraRemedios}{CegarraRemedios, theorem 9}).

\hypertarget{remark_on_notation}{}\paragraph*{{Remark on notation}}\label{remark_on_notation}

There are two uses of $\bar W$ in this area, one is as used in (\hyperlink{CegarraRemedios}{CegarraRemedios}) where it is used for the codiagonal (denoted ``$\nabla$'' above), the other is for the [[classifying space]] functor for a [[simplicial group]]. This latter is not only the older of the two uses, but also comes with a related $W$ construction. The relationship between the two is that given a simplicial group or simplicially enriched groupoid, $G$, applying the [[nerve]] functor in each dimension gives a bisimplicial set and $\bar{W}G = \nabla Ner G$. Because of this, some care is needed when using these sources.

\hypertarget{bisimplicial_abelian_groups}{}\subsection*{{Bisimplicial abelian groups}}\label{bisimplicial_abelian_groups}

\begin{prop}
\label{}\hypertarget{}{}
Let $A,B : \Delta^{op} \times \Delta^{op} \to Ab$ be bisimplicial abelian groups. A morphism $f : A \to B$ which is degreewise in one argument a weak equivalence $f_{n,\bullet} : A(n,\bullet) \to B(n,\bullet)$ induces a weak equivalence $d(f) : d(A) \to d(B)$ of the associated diagonal complexes.

\end{prop}
\begin{proof}
This is Lemma 2.7 in \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi}{chapter 4} of (\hyperlink{GoerssJardine}{GoerssJardine})

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

Some standard material is for instance in

\begin{itemize}%
\item [[Paul Goerss]] and [[Rick Jardine]], \emph{Simplicial Homotopy Theory}

\end{itemize}
\begin{itemize}%
\item [[Rick Jardine]], Lecture 008 (2010) (\href{http://www.math.uwo.ca/~jardine/papers/HomTh/lecture008.pdf}{pdf})


\item [[Samuel Isaacson]], \emph{Excercises in homotopy colimits} (\href{http://www-math.mit.edu/~mbehrens/TAGS/Isaacson_exer.pdf}{pdf})



\end{itemize}
The total simplicial set functor goes back to

\begin{itemize}%
\item [[M. Artin]], [[B. Mazur]], \emph{On the Van Kampen theorem} , Topology 5 (1966) 179--189.

\end{itemize}
The diagonal, total d\'e{}calage and total simplicial set constructions are discussed in

\begin{itemize}%
\item [[Antonio Cegarra]], [[Josué Remedios]], \emph{The relationship between the diagonal and the bar constructions on a bisimplicial set}, Topology and its applications, volume 153 (1) (2005) (\href{http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf}{pdf})


\item [[Antonio Cegarra]], [[Josué Remedios]], \emph{The behaviour of the $\bar W$-construction on the homotopy theory of bisimplicial sets}, Manuscripta Mathematica, volume 124 (4) Springer (2007)



\end{itemize}
\begin{itemize}%
\item [[Danny Stevenson]], \emph{D\'e{}calage and Kan's simplicial loop group functor} (\href{http://arxiv.org/abs/1112.0474}{arXiv:1112.0474})

\end{itemize}
The diagonal-induced model structure on $ssSet$ is discussed in

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Bisimplicial sets and the group completion theorem} in \emph{Algebraic K-Theory: Connections with Geometry and Topology} , pp 225--240. Kluwer, Dordrecht (1989)

\end{itemize}
The behaviour of fibrations under [[geometric realization]] of bisimplicial sets is discussed in

\begin{itemize}%
\item D. Anderson, \emph{Fibrations and geometric realization} , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (\href{http://projecteuclid.org/euclid.bams/1183541139}{ProjEuclid})

\end{itemize}
Discussion of respect of $\bar W$ for fibrant objects is discussed in fact 2.8 of

\begin{itemize}%
\item [[Antonio M. Cegarra]], Benjam\'i{}n A. Heredia, Josu\'e{} Remedios, \emph{Double groupoids and homotopy 2-types} (\href{http://arxiv.org/abs/1003.3820}{arXiv:1003.3820})

\end{itemize}
[[!redirects bisimplicial set]] [[!redirects bisimplicial sets]]

[[!redirects simplicial simplicial set]] [[!redirects simplicial simplicial sets]]

[[!redirects total simplicial set]] [[!redirects total simplicial sets]]

[[!redirects total décalage]]



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