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\begin{document}

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\section*{decalage}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DefinitionInComponents}{In components}\dotfill \pageref*{DefinitionInComponents} \linebreak
\noindent\hyperlink{AsARestrictionOfTotalDecalage}{As a restriction of total d\'e{}calage}\dotfill \pageref*{AsARestrictionOfTotalDecalage} \linebreak
\noindent\hyperlink{DefinitionInTermsOfCones}{In terms of cones}\dotfill \pageref*{DefinitionInTermsOfCones} \linebreak
\noindent\hyperlink{MorphismsOutOfIt}{Morphisms out of the d\'e{}calage}\dotfill \pageref*{MorphismsOutOfIt} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{FibrationResolution}{Fibration resolution}\dotfill \pageref*{FibrationResolution} \linebreak
\noindent\hyperlink{DecalageComonad}{D\'e{}calage comonad}\dotfill \pageref*{DecalageComonad} \linebreak
\noindent\hyperlink{total_dcalage}{Total D\'e{}calage}\dotfill \pageref*{total_dcalage} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_simplicial_groups}{For simplicial groups}\dotfill \pageref*{for_simplicial_groups} \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}

If you take a simplicial set and `throw away' the last face and degeneracy, and relabel, shifting everything down one `notch', you get a new simplicial set. This is what is called the \emph{d\'e{}calage} of a [[simplicial set]].

It is a model for the [[path space object]] of $X$, --or rather: the union of all based path space objects for all basepoints $x \in X_0$ -- similar to, but a little smaller than, the model $X^I \times_X X_0$, which is discussed for instance at \emph{[[factorization lemma]]}:

In the latter case an $n$-cell in the path space is a morphism to $X$ from the simplicial cone over the $n$-simplex modeled as the [[pushout]] $(\Delta[n] \times \Delta[1]) \coprod_{\Delta[n]} \Delta[0]$. This is the simplicial set obtained by forming the simplicial cylinder over $\Delta[n]$ and then contracting one end to the point.

Contrary to that, an $n$-simplex in the d\'e{}calage of $X$ is a morphism to $X$ from the cone over $\Delta[n]$ modeled simply by the [[join of simplicial sets]] $\Delta[n] \star \Delta[0]$.

This is a much smaller model for the cone. In fact $\Delta[n]\star \Delta[0] = \Delta[n+1]$ is just the $(n+1)$-simplex. On the other hand, the above pushout-construction produces simplicial sets with many $(n+1)$-simplices, the one that one ``expects'', but glued to others with some degenerate edges. Accordingly, there is, for $n \geq 1$, a proper inclusion

\begin{displaymath}
\Delta[n] \star \Delta[1] \hookrightarrow
  (\Delta[n] \times \Delta[1]) \coprod_{\Delta[n]} \Delta[0]
  \,.
\end{displaymath}
As a result, the d\'e{}calage construction is often more convenient than forming $X^I \times_X X_0$.

A central application is the special case where $X = \bar W G$ is the simplicial delooping of a [[simplicial group]] $G$ (see at \emph{[[simplicial principal bundle]]}). In this case $Dec_0 \bar W G$, called $W G$, is a standard model for the \emph{\href{simplicial%20principal%20bundle#UniversalSimplicialBundle}{universal simplicial principal bundle}}.

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

The plain definition of the d\'e{}calage of a simplicial set is very simple, stated below in

\begin{itemize}%
\item \emph{\hyperlink{DefinitionInComponents}{In components}}

\end{itemize}
However, in order to appreciate and handle this definition, it is useful to understand it as a special case of total d\'e{}calage, stated below in

\begin{itemize}%
\item \emph{\hyperlink{AsARestrictionOfTotalDecalage}{As a restriction of total d\'e{}calage}} .

\end{itemize}
From this one sees more manifestly that the d\'e{}calage of a simplicial set is built from cones in the original simplicial set. This we discuss below in

\begin{itemize}%
\item \emph{\hyperlink{DefinitionInTermsOfCones}{In terms of cones}}.

\end{itemize}
In this last formulation it is clearest what the two canonical morphisms out of the d\'e{}calage of a simplicial set mean. These we define in

\begin{itemize}%
\item \emph{\hyperlink{MorphismsOutOfIt}{Morphisms out of d\'e{}calage}}.

\end{itemize}
\hypertarget{DefinitionInComponents}{}\subsubsection*{{In components}}\label{DefinitionInComponents}

Concretely, the d\'e{}calage construction is the following.

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[simplicial set]], the \textbf{d\'e{}calage} $Dec_0\, X \in sSet$ of $X$, is the simplicial set obtained by shifting every dimension down by one, `forgetting' the last face and degeneracy of $X$ in each dimension:

\begin{itemize}%
\item $(Dec_0 \, X)_n := X_{n+1}$;
\item $d_k^{n,Dec_0 X}  := d^{n+1,X}_{k}$;
\item $s_k^{n,Dec_0 X}  := s^{n+1,X}_{k}$.

\end{itemize}
\end{defn}
\hypertarget{AsARestrictionOfTotalDecalage}{}\subsubsection*{{As a restriction of total d\'e{}calage}}\label{AsARestrictionOfTotalDecalage}

It is often useful to understand this as a special case of the \emph{[[total décalage]]} construction:

\begin{defn}
\label{}\hypertarget{}{}
Write $\sigma : \Delta_a \times \Delta_a \to \Delta_a$ for the [[ordinal sum]] operation on the [[asSet|augmented simplex category]]. The [[total décalage]] functor is precompositon with this

\begin{displaymath}
\sigma^* : sSet_a \to ssSet_a
\end{displaymath}
or rather its restriction from [[augmented simplicial sets]] to just [[simplicial sets]]/[[bisimplicial sets]].

\begin{displaymath}
\sigma^* : sSet \to ssSet
  \,.
\end{displaymath}
In terms of this the plain [[décalage]] is the functor induced from the restriction $\sigma(-,[0]) : \Delta \to \Delta$, of [[ordinal sum]] with $0$, i.e.

\begin{displaymath}
Dec_0 X := (\sigma(-,[0]))^* X
  \,.
\end{displaymath}
\end{defn}
\hypertarget{DefinitionInTermsOfCones}{}\subsubsection*{{In terms of cones}}\label{DefinitionInTermsOfCones}

The perspective from total d\'e{}calage makes fairly manifest that d\'e{}calage forms [[cones]] in $X$, as we discuss now. To this end, notice the relation of [[total décalage]] to [[join of simplicial sets]]:

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\Box : sSet \times sSet \to ssSet
\end{displaymath}
for the \emph{box product functor} that takes $X,Y \in sSet$ to the [[bisimplicial set]]

\begin{displaymath}
(X \Box Y) : ([k],[l]) \mapsto X_k \times X_l
  \,.
\end{displaymath}
\end{defn}
\begin{lemma}
\label{}\hypertarget{}{}
If $X, Y \in$ [[sSet]] are connected, then their [[join of simplicial sets]] $X \star Y$ is expressed by the [[left adjoint]] to [[total décalage]] as

\begin{displaymath}
\sigma_!(X \Box Y) = X \star Y
  \,.
\end{displaymath}
\end{lemma}
This appears as (\hyperlink{Stevenson12}{Stevenson 12, lemma 2.1}).

It follows that the left adjoint of plain d\'e{}calage forms joins with the 0-simplex:

\begin{cor}
\label{}\hypertarget{}{}
The [[left adjoint]] to $Dec_0 : sSet \to sSet$ is

\begin{displaymath}
C := \sigma_!((-) \Box \Delta[0])
  \,.
\end{displaymath}
In particular for $S \in sSet$ connected we have

\begin{displaymath}
C(S) = S \star \Delta[0]
  \,.
\end{displaymath}
\end{cor}
This appears as (\hyperlink{Stevenson12}{Stevenson 12, cor. 2.1}).

\begin{remark}
\label{}\hypertarget{}{}
The [[join of simplicial sets]] with the 0-simplex $X \star \Delta[0]$ forms a simplicial model for the [[cone]] over $X$.

\end{remark}
\begin{cor}
\label{InTermsOfCones}\hypertarget{InTermsOfCones}{}
By [[adjunction]] we have for all $n \in \mathbb{N}$

\begin{displaymath}
(Dec_0 X)_n = Hom_{sSet}( \Delta[n] \star \Delta[0], X)
  \,.
\end{displaymath}
\end{cor}
So this exhibits the $n$-cells of $Dec_0 X$ as being the cones of $n$-simplices in $X$.

\hypertarget{MorphismsOutOfIt}{}\subsubsection*{{Morphisms out of the d\'e{}calage}}\label{MorphismsOutOfIt}

\begin{prop}
\label{TheCanonicalMorphisms}\hypertarget{TheCanonicalMorphisms}{}
For $X \in sSet$ its d\'e{}calage $Dec_0 X$ comes with two canonical morphisms out of it

\begin{displaymath}
\itexarray{
    Dec_0 X &\to& X
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    const X_0
  }
  \,.
\end{displaymath}
Here in terms of the description \hyperlink{DefinitionInTermsOfCones}{above} of d\'e{}calage by cones:

\begin{itemize}%
\item the horizontal morphism is induced from the canonical inclusion $\Delta[n] \hookrightarrow \Delta[n]\star \Delta[0]$;


\item the vertical morphism is given by the canonical inclusion $\Delta[0] \hookrightarrow \Delta[n]\star \Delta[0]$.



\end{itemize}
Or in terms of components, as discussed \hyperlink{DefinitionInComponents}{above},

\begin{itemize}%
\item the horizontal morphism is given by $d_{last} : Dec_0 Y \to Y$, hence in degree $n$ by the remaining face map $d_{n+1} : X_{n+1} \to X_n$;


\item the vertical morphism is given in degree 0 by $s_0 : X_1 \to X_0$ and in every higher degree similarly by $s_0 \circ s_0 \circ \cdots \circ s_0$.



\end{itemize}
\end{prop}
See for instance (\hyperlink{Stevenson11}{Stevenson, around def. 2}) for an account.

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

\hypertarget{FibrationResolution}{}\subsubsection*{{Fibration resolution}}\label{FibrationResolution}

We discuss here how $Dec_0 X \to X$ is a [[resolution]] of $const X_0 \to X$ by a [[Kan fibration]].

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a simplicial set, the two morphisms from prop. \ref{TheCanonicalMorphisms} have the following properties.

\begin{itemize}%
\item the morphism $d_0 : Dec_0 X \to const X_0$ is a [[weak homotopy equivalence]], in fact a [[deformation retract]]; a weak inverse is given by the morphism which in degree 0 is the degeneracy $s_0 : X_0 \to X_1$, and so on.

\end{itemize}
If $X$ is a [[Kan complex]], then

\begin{itemize}%
\item the morphism $d_{last} : Dec_0 X \to X$ is a [[Kan fibration]];

\end{itemize}
\end{prop}
\begin{proof}
The first statement is classical, it appears for instance as (\hyperlink{Stevenson11}{Stevenson 11, lemma 5}).

For the second, notice that by remark \ref{InTermsOfCones} the lifting problem

\begin{displaymath}
\itexarray{
    \Lambda^n[n] &\to& Dec_0 X
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] &\to& X
  }
\end{displaymath}
is equivalent to the lifting problem

\begin{displaymath}
\itexarray{
    (\Lambda^n[n] \star \Delta[0]) \coprod_{\Lambda^i[n]} \Delta[n] &\to& X
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] \star \Delta[0] &\to& *
  }
  \,.
\end{displaymath}
Here the left morphism is an [[anodyne morphism]], in fact is an $(n+1)$-[[horn]] inclusion $\Lambda[n+1] \to \Delta[n+1]$. So a lift exists if $X$ is a [[Kan complex]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
By the above, $Dec_0 X$ is the [[disjoint union]] of [[over quasi-categories]]

\begin{displaymath}
Dec_0 X  = \coprod_{x \in X_0} X_{/x}
  \,.
\end{displaymath}
For each of these the statement that the projection $X_{/x} \to X$ is a [[Kan fibration]] if $X$ is a [[Kan complex]], and moreover that it is a a [[right fibration]] if $X$ is a [[quasi-category]], is (\hyperlink{Joyal}{Joyal, theorem 3.19}), reproduced also as ([[Higher Topos Theory|HTT, prop. 2.1.2.1]]). Notice that left/right fibrations into a Kan complex are automatically Kan fibrations (by the discussion at \emph{\href{right%2Fleft+Kan+fibration#AsFibrationsInInfinityGroupoids}{Left fibration in ∞-groupoids}}).

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
For $X$ a [[Kan complex]], the d\'e{}calage morphism $Dec_0 X \to X$ is a [[Kan fibration]] [[resolution]] of the inclusion $const X_0 \to X$ of the set of 0-cells of $X$, regarded as a [[discrete object|discrete]] simplicial set:

there is a diagram

\begin{displaymath}
\itexarray{
    const X_0 &\stackrel{\simeq}{\to}& Dec_0 X
    \\
    \downarrow && \downarrow
    \\
    X &\to& X
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item the top morphism

\begin{itemize}%
\item is given in degree $n$ by the $n$-fold degeneracy map $s_0 \circ s_0 \circ \cdots s_0$;


\item is a [[weak homotopy equivalence]];



\end{itemize}

\item the right vertical morphism

\begin{itemize}%
\item is given in degree $n$ by $d_{n+1} : X_{n+1} \to X_n$


\item is a [[Kan fibration]].



\end{itemize}


\end{itemize}
\end{cor}
\begin{remark}
\label{}\hypertarget{}{}
The inclusion $const X_0 \to X$ presents a canonical [[effective epimorphism in an (∞,1)-category]] in [[∞Grpd]] into $X$, out of a [[0-truncated]] object. By the above, the d\'e{}calage is a natural fibration resolution of this canonical ``[[atlas]]''.

This is useful for instance in the discussion of [[homotopy pullbacks]] of this effective epimorphism: by the discussion there the homotopy pullback of $const X_0 \to X$ along any morphism $f : A \to X$ is presented by the ordinary pullback of any Kan fibration resolution, hence in particular of the d\'e{}calage projection:

\begin{displaymath}
f^* Dec_0 X \simeq A \times_X^{h} const X_0
  \,.
\end{displaymath}
\end{remark}
\hypertarget{DecalageComonad}{}\subsubsection*{{D\'e{}calage comonad}}\label{DecalageComonad}

D\'e{}calage also has an abstract [[category theory|category theoretic]] description as follows. The [[simplex category]], as a [[monoidal category]] $(\Delta, +, 0)$ equipped with the [[monoid]] $1$, is the ``[[walking structure|walking]] [[monoid]]'', i.e., is initial among monoidal categories equipped with a monoid. Therefore $\Delta^{op}$ is the walking [[comonoid]]; as a result, there is a [[comonad]]

\begin{displaymath}
- + 1: \Delta^{op} \to \Delta^{op}
\end{displaymath}
which induces a comonad on simplicial sets whose underlying functor is precisely d\'e{}calage:

\begin{displaymath}
Dec: Set^{\Delta^{op}} \to Set^{\Delta^{op}}
\end{displaymath}
The map $d_{last}: Dec_0 \to Id$ is the counit of this comonad. The comonad itself is analogous to a kind of unbased [[path space object]] comonad $P$ on $Top$ whose value at a space $X$ is a pullback

\begin{displaymath}
\itexarray{
P X & \to & X^I \\
\downarrow & & \downarrow eval_0 \\
|X| & \stackrel{i}{\to} & X
}
\end{displaymath}
where $i$ is the set-theoretic identity inclusion of $X$ equipped with the discrete topology. Thus we have

\begin{displaymath}
P X = \sum_{x_0 \in X} P(X, x_0),
\end{displaymath}
the sum over all possible basepoints $x_0$ of path spaces based at $x_0$. The analogy is made precise by a canonical isomorphism

\begin{displaymath}
Dec_0 \circ S \cong S \circ P
\end{displaymath}
where $S: Top \to Set^{\Delta^{op}}$ is simplicial singularization.

A $P$-coalgebra partitions $X$ into path components and exhibits contractibility of each component. Similarly, a coalgebra of the decelage comonad exhibits the acyclicity of the underlying simplicial set.

\hypertarget{total_dcalage}{}\paragraph*{{Total D\'e{}calage}}\label{total_dcalage}

Using either the simplicial [[comonadic resolution]] generated by the above comonad or directly using [[ordinal sum]], we get a [[bisimplicial set]] known as the [[total decalage|total décalage]] of $Y$. See there for more details.

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

\hypertarget{for_simplicial_groups}{}\subsubsection*{{For simplicial groups}}\label{for_simplicial_groups}

The case of $Dec_0 G$ for $G$ a [[simplicial group]] is important in the simplicial theory of algebraic models for [[homotopy n-types]].

In this case the morphism $d_{last} : Dec_0\, G \to G$, is an [[epimorphism]]. Taking the [[kernel]] of this and then applying $\pi_0$, yields a [[crossed module]] constructed from the [[Moore complex]] of $G$

\begin{displaymath}
N G_1/d_2(NG_2)\to N G_0,
\end{displaymath}
which has kernel $\pi_1(G)$ and cokernel $\pi_0(G)$. This crossed module represents the [[homotopy 2-type]] of $G$. Applying the d\'e{}calage twice leads to a [[crossed square]] which represents the 3-type of $G$, \ldots{} and so on.

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

\begin{itemize}%
\item [[bar construction]]

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

Original sources are

\begin{itemize}%
\item [[Luc Illusie]], \emph{Complexe cotangent et d\'e{}formations I}, volume 239 of Lecture Notes in Maths , Springer-Verlag. and 1972, \emph{Complexe cotangent et d\'e{}formations II}, volume 283 of Lecture Notes in Maths , Springer-Verlag (1971)

\end{itemize}
and

\begin{itemize}%
\item [[John Duskin]], \emph{Simplicial methods and the interpretation of ``triple'' cohomology}, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc (1975)

\end{itemize}
The notion of d\'e{}calage has been widely used since the paper introducing the method of [[cohomological descent]] in [[Hodge theory]]:

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge. III}, Inst. Hautes \'E{}tudes Sci. Publ. Math. \textbf{44} (1974), 5--77.

\end{itemize}
Reviews are in

\begin{itemize}%
\item [[Phil Ehlers]], \emph{Algebraic Homotopy in Simplicially Enriched Groupoids}, 1993, University of Wales Bangor, (pdf [[Ehlers-thesis.pdf|here:file]])

\end{itemize}
The link with simplicial groups and algebraic models of homotopy $n$-types is given in

\begin{itemize}%
\item [[Tim Porter]], \emph{n-types of simplicial groups and crossed n-cubes}, Topology, 32, (1993), 5--24.


\item [[Tim Porter]], \emph{[[Menagerie|The crossed menagerie]]}



\end{itemize}
A detailed account of various technical aspects is in

\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}
and in section 2.2 of

\begin{itemize}%
\item [[Danny Stevenson]], \emph{Classifying theory for simplicial parametrized groups} (\href{http://arxiv.org/abs/1203.2461}{arXiv:1203.2461})

\end{itemize}
Closely related technical results are in section 3 of

\begin{itemize}%
\item [[André Joyal]], \emph{The theory of quasi-categories and its applications} , lectures at CRM Barcelona (2008)

\end{itemize}
An application in the theory of [[stacks]] is discussed in

\begin{itemize}%
\item [[Anders Kock]], \emph{The stack quotient of a groupoid}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, \textbf{44} no. 2 (2003), p. 85--104 \href{http://www.numdam.org/item?id=CTGDC_2003__44_2_85_0}{numdam}

\end{itemize}
[[!redirects decalage]] [[!redirects décalage]]



\end{document}