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\begin{document}

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\section*{generalized Reedy model structure}

\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{the_model_structure}{The model structure}\dotfill \pageref*{the_model_structure} \linebreak
\noindent\hyperlink{GlobalLatchingObjects}{Degreewise latching and matching objects}\dotfill \pageref*{GlobalLatchingObjects} \linebreak
\noindent\hyperlink{SkeletaAndCoskeleta}{Skeleta and coskeleta}\dotfill \pageref*{SkeletaAndCoskeleta} \linebreak
\noindent\hyperlink{proof_of_the_model_category_axioms}{Proof of the model category axioms}\dotfill \pageref*{proof_of_the_model_category_axioms} \linebreak
\noindent\hyperlink{Lifting}{Lifting}\dotfill \pageref*{Lifting} \linebreak
\noindent\hyperlink{Factorization}{Factorization}\dotfill \pageref*{Factorization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{HomotopySkeletalFiltration}{Homotopy skeletal filtration and coskeleton tower}\dotfill \pageref*{HomotopySkeletalFiltration} \linebreak
\noindent\hyperlink{relation_to_other_model_structures}{Relation to other model structures}\dotfill \pageref*{relation_to_other_model_structures} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \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}

Generalized Reedy model structures are a class of [[model categories]] that generalize the [[Reedy model structures]] when the underlying [[site]] is generalized from a [[Reedy category]] to a [[generalized Reedy category]].

So these model structures serve to present [[(∞,1)-categories of (∞,1)-functors]] on generalized Reedy categories.

As for the [[Reedy model structure|Reedy model structures]], the generalized Reedy model structure typically models [[geometric shapes for higher structures]]. The crucial generalization is that the basic shapes here may have non-trivial [[automorphisms]].

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

Let $S$ be a (Berger-Moerdijk) [[generalized Reedy category]]. Let $\mathcal{C}$ be a category with small [[limits]] and [[colimits]].

\hypertarget{the_model_structure}{}\subsubsection*{{The model structure}}\label{the_model_structure}

\begin{defn}
\label{FirstNotation}\hypertarget{FirstNotation}{}
For every [[object]] $s \in S$, every [[functor]] $X : S \to \mathcal{C}$ and every [[natural transformation]] $\phi : X \to Y$

\begin{itemize}%
\item $S^+(s)$ for the [[full subcategory]] of the [[slice category]] $S^+/s$ on the non-invertible morphisms into $s$;


\item $S^-(s)$ for the [[full subcategory]] of the [[under category]] $s/S^-$ on the non-invertible morphisms out of $s$;


\item write

\begin{displaymath}
Latch_s(X) := {\lim_{\underset{(r \to s) \in S^+(s)}{\to}}} X(r)
\end{displaymath}
for the [[colimit]] of $X$ over $S^+(s)$, called the \textbf{latching object} of $X$ at $s$;


\item write

\begin{displaymath}
Match_s(X) := {\lim_{\underset{(s \to r) \in S^-(s)}{\leftarrow}}} X(r)
\end{displaymath}
for the [[limit]] of $X$ over $S^-(s)$, called the \textbf{matching object} of $X$ at $s$.


\item write

\begin{displaymath}
X_s \coprod_{Latch_s(X)} Latch_s(Y) \to Y_s
\end{displaymath}
for the universal morphism induced from the morphism $L_s(X) \to X_s$, called the \textbf{relative latching map} of $\phi$ at $s$;


\item write

\begin{displaymath}
X_s \to Match_s(X) \prod_{Match_s(Y)} Y_s
\end{displaymath}
for the dual universal morphism, called the \textbf{relative matching map} of $\phi$ at $s$.



\end{itemize}
\end{defn}
See \emph{[[Joyal-Tierney calculus]]} for more on these kinds of objects and morphisms.

\begin{remark}
\label{}\hypertarget{}{}
In the above situation, the [[automorphism]] group $Aut_S(s)$ of $s$ canonically acts on all objects that appear, and all morphisms that appear respect this action.

Equivalently this means that for all $s$ the above objects and morphisms take place in the presheaf category $[B Aut(s), \mathcal{C}]$.

\end{remark}
\begin{proof}
Since limits and colimits in [[presheaf categories]] are computed objectwise.

\end{proof}
Choose now once and for all on all on $[\coprod_{s \in S} B Aut(s), \mathcal{C}]$ either the projective or the injective [[model structure on functors]] (if they exist). We will use the subscript ``${}_{proj/inj}$'', as in $[\coprod_{s \in S} B Aut(s), \mathcal{C}]_{proj/inj}$, to indicate some fixed choice.

\begin{defn}
\label{ReedyModelStructure}\hypertarget{ReedyModelStructure}{}
Let $S$ be a (Berger-Moerdijk)-[[generalized Reedy category]].

And $\mathcal{C}$ be a [[model category]] such that the [[model structure on functors]] $[\coprod_{s \in S} B Aut(s), \mathcal{C}]_{proj/inj}$ exists (sufficient for the projective structure is that $\mathcal{C}$ is a [[cofibrantly generated model category]] and sufficient for the injective structure is that it is a [[combinatorial model category]]).

Write $[S, \mathcal{C}]$ for the [[category of presheaves]] on $S^{op}$ with values in $\mathcal{C}$.

Call a morphism $f : X \to Y$ in $[S, \mathcal{C}]$

\begin{itemize}%
\item a \textbf{Reedy cofibration} if for each $s \in S$ the relative latching map

\begin{displaymath}
X_s \coprod_{L_s(X)} L_s(Y) \to Y_s
\end{displaymath}
is a cofibration in $[B Aut(s), \mathcal{C}]_{proj/inj}$;


\item a \textbf{Reedy fibration} if for each $s \in S$ the relative matching map

\begin{displaymath}
X_s \to M_s(X) \prod_{M_s(Y)} Y_s
\end{displaymath}
is a fibration in $[S, \mathcal{C}]_{proj/inj}$;


\item a \textbf{Reedy weak equivalence} if for each $s \in S$ the morphism

\begin{displaymath}
f_s : X_s \to Y_s
\end{displaymath}
is a weak equivalence in $\mathcal{C}$.



\end{itemize}
\end{defn}
\hypertarget{GlobalLatchingObjects}{}\subsubsection*{{Degreewise latching and matching objects}}\label{GlobalLatchingObjects}

We discuss here an alternative way of speaking about the latching and matching objects, one where all indices at a given \emph{degree} $n \in \mathbb{N}$ are collected.

Recall from def. \ref{FirstNotation} that for $s \in S$ we write $S^+(s)$ for the category of non-invertible degree-increasing morphisms into $s$. We introduce the union of these categories over all objects of a fixed degree.

\begin{defn}
\label{SecondNotation}\hypertarget{SecondNotation}{}
For $n \in \mathbb{N}$ write

\begin{itemize}%
\item $S^+(n) = \coprod_{s \in S \atop d(s) = n} S^+(s)$;


\item $d_ n : S^+((n)) \to S$ for the restriction of the [[domain opfibration]] to objects that are non-invertible morphisms in $S^+$ with codomain in degree $n$ and to morphisms whose codomain is invertible, i.e. to diagrams of the form

\begin{displaymath}
\itexarray{
     a &\stackrel{\in S^+}{\to}& b
     \\
     \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} & not\, invertible
     \\
     s & \stackrel{\simeq}{\to} & s' & deg = n
  }
  \,;
\end{displaymath}

\item $i_n : S^+(n) \hookrightarrow S^+((n))$ for the [[full subcategory]] inclusion on constant codomains;


\item $G_n(S) \subset Core(S)$ for the [[groupoid]] of objects of degree $n$ and [[isomorphisms]] between them.



\end{itemize}
\end{defn}
\begin{prop}
\label{DiagramOfRestrictions}\hypertarget{DiagramOfRestrictions}{}
The above categories and functors arrange into a [[diagram]]

\begin{displaymath}
\itexarray{
    S &\stackrel{dom_n}{\leftarrow}& S^+((n)) &\stackrel{cod_n}{\to}& G_n(S)
    &\stackrel{j_n}{\hookrightarrow}& S
    \\
    &&
    {}^{\mathllap{i_n}}\uparrow && \uparrow^{\mathrlap{i_n}}
    \\
    && S^+(n) &\stackrel{cod_n}{\to}& Obj(S)_n
  }  
  \,,
\end{displaymath}
where the vertical morphisms are (non-full) inclusions and the square is a [[pullback]] (in the 1-category [[Cat]]) of an [[opfibration]]. Therefore it satisfies the [[Beck-Chevalley condition]] (see the discussion there) so that we have a [[natural isomorphism]]

\begin{displaymath}
(cod_n)_! i_n^* \simeq i_n^* (cod_n)_!
  \,,
\end{displaymath}
where $(cod_n)_!$ denotes left [[Kan extension]] along $cod_n$.

\end{prop}
\begin{remark}
\label{RestrictedCodomainIsOpfibration}\hypertarget{RestrictedCodomainIsOpfibration}{}
The restricted [[codomain fibration|codomain opfibration]] $cod_n : S^+((n)) \to G_n$ is indeed still an [[opfibration]]: it is the [[Grothendieck construction]] of the [[pseudofunctor]]

\begin{displaymath}
S^+(-) : G_n(S) \to Cat
\end{displaymath}
\begin{displaymath}
s \mapsto S^+(s)
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{GlobalLatching}\hypertarget{GlobalLatching}{}
For $n \in \mathbb{N}$, let $X \in [S, \mathcal{C}]$. Write

\begin{itemize}%
\item $X_n := j_n^* X = X|_{G_n(S)} \in [G_n(S), \mathcal{C}]$;


\item the \textbf{$n$th latching object} is

\begin{displaymath}
Latch_n(X) := (cod_n)! dom_n^* X \in [G_n(S), \mathcal{C}]
  \,.
\end{displaymath}

\item the \textbf{$n$th latching morphism}

\begin{displaymath}
Latch_n(X) \to X_n
\end{displaymath}
is the [[adjunct]] to the canonical functor

\begin{displaymath}
dom_n^* X \to (j_n cod_n)^* X
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
The following proposition says that the ``global'' latching objects indeed contain all the ordinary latching objects in the given degree.

\begin{prop}
\label{GlobalLatchingContainsLocalLatchings}\hypertarget{GlobalLatchingContainsLocalLatchings}{}
For $s \in Obj(S)_n$ we have

\begin{displaymath}
Latch_n(X) : s \mapsto Latch_s(X)
\end{displaymath}
and the component of the $n$th latching morphism on $s$ is the canonical $Latch_s(X) \to X(s)$.

\end{prop}
\begin{proof}
By remark \ref{RestrictedCodomainIsOpfibration} $(cod_n)_!$ is the left [[Kan extension]] along an [[opfibration]]. By a standard fact (see \href{http://ncatlab.org/nlab/show/Kan+extension#AlongFibrations}{here} at \emph{[[Kan extension]]}) these are computed at any object by the colimit over the fiber over that object.

By definition, that fiber is

\begin{displaymath}
cod_n^{-1}(s)
  = 
  \left\{
    \itexarray{
      a &&\stackrel{\in S^+}{\to}&& b
      \\
      & \searrow^{\mathrlap{\in S^+}} && \swarrow_{\mathrlap{\in S^+}}  && non\; invertible
      \\
      && s
    }
  \right\}
  \,.
\end{displaymath}
This is indeed $S^+(s)$ (by the essential uniqueness of the $S^+\circ S^-$-factorization, this necessarily has the morphisms $a \to b$ in $S^+$, too.)

So

\begin{displaymath}
\begin{aligned}
  ((cod_n)_! dom_n^* X)(s)
  & \simeq
  \lim_{\underset{S^+(s)}{\to}} (X \circ dom)
  \\
  & \simeq
  =: Latch_s(X)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
An entirely dual discussion gives the degreewise matching objects: we have a diagram of categories

\begin{displaymath}
\itexarray{
    S &\stackrel{cod_n}{\leftarrow}& S^-((n)) &\stackrel{dom_n}{\to}& G_n(S)
    &\stackrel{j_n}{\hookrightarrow}& S
    \\
    &&
    {}^{\mathllap{i_n}}\uparrow && \uparrow^{\mathrlap{i_n}}
    \\
    && S^+(n) &\stackrel{dom_n}{\to}& Obj(S)_n
  }  
  \,,
\end{displaymath}
and

\begin{displaymath}
Match_n X := (dom_n)_* cod_n^* X
  \,,
\end{displaymath}
where $(dom_n)_*$ is the right [[Kan extension]] along $dom_n$.

\hypertarget{SkeletaAndCoskeleta}{}\subsubsection*{{Skeleta and coskeleta}}\label{SkeletaAndCoskeleta}

Over any generalized Reedy category there is an anlog of the notion of [[simplicial skeleton]] and [[simplicial coskeleton]].

For $n \in \mathbb{N}$, write

\begin{displaymath}
t_n : S_{\leq n}  \hookrightarrow S
\end{displaymath}
for the [[full subcategory]] on the objects of degree $\leq n$.

\begin{defn}
\label{SkeletaByAdjunction}\hypertarget{SkeletaByAdjunction}{}
Left and right [[Kan extension]] along $t_n$ defines an [[adjoint triple]]

\begin{displaymath}
((t_n)_! \dashv t_n^* \dashv (t_n)_*)
  :
   [S_{\leq n},\mathcal{C}] \stackrel{\overset{(t_n)_!}{\hookrightarrow}}{\stackrel{\overset{t_n^*}{\leftarrow}}{\underset{(t_n)_*}{\hookrightarrow}}}
  [S,\mathcal{C}]
  \,.
\end{displaymath}
The induced [[monads]]

\begin{displaymath}
(sk_n \dashv cosk_n)
  : 
  [S,\mathcal{C}]
    \stackrel{\overset{(t_n)_!}{\leftarrow}}{\underset{t_n^*}{\to}}
  [S_{\leq n},\mathcal{C}]
    \stackrel{\overset{t_n^*}{\leftarrow}}{\underset{(t_n)_*}{\to}}
  [S,\mathcal{C}]
\end{displaymath}
are the \textbf{$n$-[[skeleton]]} and \textbf{$n$-[[coskeleton]]} functors, respectively.

Define for all $X \in [S, \mathcal{C}]$ the notation $sk_{-1}X$ to denote the [[initial object]] and $cosk_{-1}X$ the [[terminal object]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Here $(t_n)_!$ and $(t_n)_*$ are indeed [[full and faithful functors]], as indicated.

\end{remark}
\begin{proof}
Since $t_{n-1}$ is a [[full and faithful functor]], so is its left Kan extension (see \href{Kan+extension#LeftKanOnRepresentables}{here} at \emph{[[Kan extension]]}). Moreover in an [[adjoint triple]] the leftmost functor is full and faithful if and only if the rightmost one is.

\end{proof}
The $((t_n)_! \dashv t_n^*)$-[[unit of an adjunction|counit]] and the $(t_n^* \dashv (t_n)_*)$-unit induces [[natural transformations]]

\begin{displaymath}
sk_n \to id
\end{displaymath}
\begin{displaymath}
id \to cosk_n
  \,.
\end{displaymath}
\begin{lemma}
\label{LatchingIsSkeleton}\hypertarget{LatchingIsSkeleton}{}
For all $n \in \mathbb{N}$, the $n$th latching object, def. \ref{GlobalLatching}, is isomorphic to the $(n-1)$-skeleton in degree $n$, and dually, the degree-$n$ matching object is isomorphic to the $(n-1)$-coskeleton in degree $n$. Under these identifications the canonical morphisms on both sides match

\begin{displaymath}
Latch_n(X) \simeq (sk_{n-1}(X))_n
\end{displaymath}
\begin{displaymath}
Match_n(X) \simeq (cosk_{n-1}(X))_n
  \,.
\end{displaymath}
\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 6.2}).

\begin{proof}
Observe that for any $s \in S$ of degree $n$, the canonical inclusion

\begin{displaymath}
i_s : S^+(s) \hookrightarrow t_{n-1}/ s
\end{displaymath}
into the [[comma category]] is a [[cofinal functor]]:

\begin{itemize}%
\item for $d \to s$ any object in $t_{n-1}/s$ it factors essentially uniquely as $d \stackrel{\in S^-}{\to} \stackrel{\in S^+}{\to} s$, and hence the [[comma category ]] $d/i_s$ is non-empty;


\item similarly, since every morphism factors essentially uniquely in $S$, there is a zig-zag between any two objects in $d / i_s$ constructed from the isomorphisms that exhibit the essentially unique factorization.



\end{itemize}
With this the statement follows from the fact that restriction along cofinal functors preserves colimits and the pointwise description of left [[Kan extension]] by [[colimits]] over comma categories:

\begin{displaymath}
\begin{aligned}
    sk_{n-1}(X)_n(s)
    & :=
    (j_n^* (t_{n-1})_! t_{n-1}^* X )(s)
    \\
    & \simeq 
    ((t_{n-1})_! t_{n-1}^* X )(s)
    \\
    & \simeq
    \lim_{\underset{t_{n-1}/ s }{\to}} X \circ t_{n-1}
    \\
    & \simeq 
    \lim_{\underset{S^+(s)}{\to}} X \circ dom_n
    \\
    & \simeq 
    Latch_n X
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{coSkeletonTower}\hypertarget{coSkeletonTower}{}
The tower of inclusions

\begin{displaymath}
S_{0} \hookrightarrow S_{\leq 1} \hookrightarrow \cdots \hookrightarrow S_{\leq n-1}
  \stackrel{q_{n-1}}{\hookrightarrow} S_{\leq n} \hookrightarrow \cdots
\end{displaymath}
induces towers of [[natural transformations]]

\begin{displaymath}
\emptyset \to sk_0 X \to sk_1 X \to sk_2 X \to \cdots \to X
  \,,
\end{displaymath}
and

\begin{displaymath}
X \to \cdots \to cosk_2 X \to cosk_1 X \to cosk_0 X \to *
  \,,
\end{displaymath}
that exhibit $X$ as the [[colimit]] of its skeleton tower and as the [[limit]] of its coskeleton tower.

\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 6.3}).

\begin{proof}
The morphisms in the tower come from the [[unit of an adjunction|adjunction units and counits]]: the morphism

\begin{displaymath}
sk_n X \to sk_{n+1} X
\end{displaymath}
is

\begin{displaymath}
(t_{n+1})_! (q_n)_! q_n^* t_{n+1}^*X \to  (t_{n+1})_! t_{n+1}^* X
  \,.
\end{displaymath}
Therefore a [[cocone]] under this morphism

\begin{displaymath}
\itexarray{
    sk_{n-1} X &&\to& sk_n X
    \\
    & \searrow && \swarrow
    \\
    && Y
  }
\end{displaymath}
is equivalently a diagram

\begin{displaymath}
\itexarray{
    (q_{n})_! q_{n}^* t_{n+1} X &&\to& t_{n+1}^* X
    \\
    & \searrow && \swarrow
    \\
    && t_{n+1}^* Y
  }
  \,,
\end{displaymath}
which in turn is equivalently just a morphism $t_n^* X \to t_n^* Y$. So a cocone under the whole tower is an object $Y$ equipped for each $n$ with a morphism $t_n^* X \to t_n^* Y$. Clearly $X$ itself is the [[initial object|inital]] such object.

\end{proof}
\begin{lemma}
\label{IdempotenceOfCoSkeleta}\hypertarget{IdempotenceOfCoSkeleta}{}
For every $X \in [S, \mathcal{C}]$ and for all pairs $k,l \in \mathbb{N}$ with $k \leq l$, we have [[natural isomorphisms]]

\begin{displaymath}
sk_k sk_l \simeq sk_l sk_k \simeq sk_k
\end{displaymath}
and

\begin{displaymath}
cosk_k cosk_l \simeq cosk_l cosk_k \simeq cosk_k
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
With the above notation we have $t_k = t_l \circ q_k$. Therefore for instance

\begin{displaymath}
sk_k sk_l X 
  \simeq
  (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X
  \,.
\end{displaymath}
Since the left adjoints here are [[full and faithful functors]] we have $t_l^* (t_l)_! \simeq id$ and hence

\begin{displaymath}
\cdots \simeq
  (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X
  \simeq
  sk_k X
  \,.
\end{displaymath}
Similarly for all the other cases.

\end{proof}
The following is a useful tool for inductively creating objects by adding higher degree components.

\begin{lemma}
\label{SkeletalExtension}\hypertarget{SkeletalExtension}{}
Given

\begin{itemize}%
\item an object $X_{\leq (n-1)} \in [S_{\leq (n-1)}, \mathcal{C}]$;


\item and an object $X_n \in [G_n(S), \mathcal{C}]$;



\end{itemize}
the choices of $X_{\leq n} \in [S_{\leq (n-1)}, \mathcal{C}]$ such that

\begin{itemize}%
\item $X_{\leq (n-1)} = t^*_{n-1} X_{\leq n}$


\item $X_n = j_n^* X$;



\end{itemize}
are in bijection with choices of morphisms

\begin{displaymath}
Latch_n(X_{\leq (n-1)}) \to X_n \to Match_n(X_{\leq (n-1)})
\end{displaymath}
in $[G_n(S), \mathcal{C}]$.

Accordingly, given morphisms $f_{\leq (n-1)} : X_{\leq (n-1)} \to Y_{\leq (n-1)}$ and $f_n : X_n \to Y_n$, then choice of extensions to a morphism $f_{\leq n} : X_{\leq n} \to Y_{\leq n}$ are in bijection with choices of vertical morphisms in commuting diagrams

\begin{displaymath}
\itexarray{
     Latch_n(X_{\leq (n-1)}) &\stackrel{Latch_n(f_{\leq (n-1)})}{\to}& Latch_n(Y_{\leq (n-1)})
     \\
     \downarrow && \downarrow
     \\
     X_n &\stackrel{f_n}{\to}& Y_n
     \\
     \downarrow && \downarrow
     \\
     Match_n(X_{\leq (n-1)})
     &\stackrel{Match_n(f_{\leq (n-1)})}{\to}&
     Match_n(Y_{\leq (n-1)})
  }
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
If the object exists, then the morphisms do, by the above definitions/discussion. Conversely, given these morphisms, we take $X_{\leq n} : S \to \mathcal{C}$ to be given by $X_{\leq (n-1)}$ on morphism of degree $\leq (n-1)$, to be given by $X_n$ on morphisms between objects of degree $n$, and need to define it on the degree-changing morphisms to and from objects of degree $n$. This information is provided precisely by the co-cone components of $Latch_n(X) \to X_n$ and by the cone-components of $X_n \to Match_n(X)$.

\end{proof}
\hypertarget{proof_of_the_model_category_axioms}{}\subsection*{{Proof of the model category axioms}}\label{proof_of_the_model_category_axioms}

We discuss that for $S$ a Berger-Moerdijk-[[generalized Reedy category]] and $\mathcal{C}$ a [[cofibrantly generated model category]], def. \ref{ReedyModelStructure} indeed defines a [[model category]] structure on the [[functor category]] $[S,\mathcal{C}]$.

\begin{theorem}
\label{}\hypertarget{}{}
Def. \ref{ReedyModelStructure} indeed defines a [[model category|model structure]].

\end{theorem}
\begin{proof}
It is clear that $[S,\mathcal{C}]$ has all limits and colimits (as for any [[category of presheaves]] they are computed objectwise in $\mathcal{E}$) and that the weak equivalences satisfy [[two-out-of-three]], since the weak equivalences in $\mathcal{E}$ do. Also, all three classes of morphisms are closed under [[retracts]], since, for instance, the relative latching morphism of a retract is the retract of a relative latching morphism and so the property follows with the retract-closure of the classes of morphisms in $\mathcal{E}$.

It remains to show that the relevant lifting and factorization properties hold. This we discuss in a list of lemmas below in \emph{\hyperlink{Lifting}{Lifting}} and \emph{\hyperlink{Factorization}{Factorization}}.

\end{proof}
\hypertarget{Lifting}{}\subsubsection*{{Lifting}}\label{Lifting}

We work with the ``global'' latching objects from \hyperlink{GlobalLatchingObjects}{above}.

\begin{remark}
\label{}\hypertarget{}{}
A morphism $f : X \to Y$ in $[S, \mathcal{C}]_{gReedy}$ is a Reedy cofibration precisely if for all $n \in \mathbb{N}$ the global relative latching morphism, def. \ref{GlobalLatching}

\begin{displaymath}
X_n \coprod_{Latch_n(X)} Latch_n(Y) \to Y_n
\end{displaymath}
is a cofibration in $[G_n(S), \mathcal{C}]_{proj/inj}$.

\end{remark}
\begin{proof}
The pushout in the presheaf category $[G_n(S), \mathcal{C}]$ is computed objectwise, so that the component of the $n$th relative latching morphism at $s \in S$ is the relative latching morphism at $s$, by prop. \ref{GlobalLatchingContainsLocalLatchings}.

The [[groupoid]] $G_n(S)$ is equivalent to the disjoint union $\coprod_{[r] \in \pi_0 G_n(S)} B Aut_S(s)$ of the automorphism groupoids of one representative in each isomorphism class. A morphism in $[G_n(S), \mathcal{C}]_{proj/inj}$ is a cofibration precisely if its restriction to all of the $[B Aut_{\mathcal{C}}(s), \mathcal{C}]_{proj/inj}$ is.

\end{proof}
\begin{lemma}
\label{LiftingLemma}\hypertarget{LiftingLemma}{}
In the generalized Reedy structure, def. \ref{ReedyModelStructure}, the following holds.

\begin{itemize}%
\item Acyclic Reedy cofibrations $f$ such that for all $n \in \mathbb{N}$ the morphism $Latch_n(f)$ is an objectwise acyclic cofibration have the [[left lifting property]] against fibrations;


\item Reedy cofibration have the left lifting property against acyclic Reedy fibrations $g$ with the special property that all $Latch_n(g)$ are objectwise acyclic fibrations.



\end{itemize}
\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 5.2, lemma 5.4}).

\begin{proof}
We show the first clause. The second is dual. So let $f : X \to Y$ be an acyclic cofibration with the above extra property, and let $g : Y \to X$ be a fibration. We will exhibit a lift in any commuting diagram

\begin{displaymath}
\itexarray{
    A &\stackrel{}{\to}& Y
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}}
    \\
    B &\stackrel{}{\to}& X
  }
\end{displaymath}
by stepwise constructing lifts in the skeletal filtration, lemma \ref{coSkeletonTower}.

At $n = 0$, observe that since $L_0(X) = \emptyset$ for all $X$, the fact that

\begin{displaymath}
A_0 \coprod_{L_0(A)} L_0(B) \to B_0
\end{displaymath}
is a cofibration in $[G_0, \mathcal{C}]_{proj/inj}$ by assumption, means that in fact $f_0 : A_0 \to B_0$ is an acyclic cofibration here. Similarly $Y_0 \to X_0$ is a fibration there. But $G_0(S) = S_{\leq 0}$ and so the restriction of the lifting problem along $t_0$

\begin{displaymath}
\itexarray{
    A_0 &\stackrel{}{\to}& Y_0
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}}
    \\
    B_0 &\stackrel{}{\to}& X_0
  }
\end{displaymath}
is a lifting problem in $[G_0(S), \mathcal{C}]_{proj/inj}$ of an acyclic cofibration against a fibration, and hence has a filler $\gamma_0 : B_0 \to Y_0$ there.

Now assume that a filler $\gamma_{\leq (n-1)}$ in

\begin{displaymath}
\itexarray{
    A_{\leq (n-1)} &\stackrel{}{\to}& Y_{\leq (n-1)}
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}}
    \\
    B_{\leq (n-1)} &\stackrel{}{\to}& X_{\leq (n-1)}
  }
\end{displaymath}
has been found. By lemma \ref{LatchingIsSkeleton} this induces maps $Latch_n(B) \to Latch_n (Y)$ and $Match_n(B) \to Match_n(Y)$ from which we can build the commuting diagram

\begin{equation}
\itexarray{
    A_n \coprod_{Latch_n A} Latch_n B
    &\to&
    Y_n
    \\
    \downarrow^{\mathrlap{v_n}} && \downarrow^{\mathrlap{w_n}}
    \\
    B_n &\to& X_n \times_{Match_n X} Match_n Y
  }
\label{MainLiftingDiagram}\end{equation}
in $[G_n(S), \mathcal{C}]$.

Here for instance the top horizontal morphism comes from the commutativity of the square

\begin{displaymath}
\itexarray{
    Latch_n A &\to& Latch_n B &\to& Latch_n Y
    \\
    \downarrow && && \downarrow
    \\
    A_n &\to& &\to& Y_n
  }
\end{displaymath}
by naturality of the $(sk_{n-1} \dashv t_{n-1}^*)$-counit.

We observe now that finding a lift in \eqref{MainLiftingDiagram} will complete the induction step. To see this in more detail, notice that in the top left the lift

\begin{displaymath}
\itexarray{
    A_n \coprod_{Latch_n A} Latch_n B
    &\to&
    Y_n
    \\
    \downarrow &\nearrow_{\gamma_n}& 
    \\
    B_n 
  }
\end{displaymath}
is

\begin{itemize}%
\item a lift in

\begin{displaymath}
\itexarray{
     A_n &\to& Y_n
     \\
     \downarrow & \nearrow_{\mathrlap{\gamma_n}}
     \\
     B_n
  }
\end{displaymath}

\item such that it makes

\begin{displaymath}
\itexarray{
    Latch_n(B) &\to& Latch_n(Y)
    \\  
    \downarrow && \downarrow
    \\
    B_n &\stackrel{\gamma_n}{\to}& Y_n
  }
\end{displaymath}
commute;



\end{itemize}
and in the bottom right the lift

\begin{displaymath}
\itexarray{
     && Y_n
     \\
     & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow
     \\
     B_n &\to& X_n \prod_{Match_n X} Match_n Y
  }
\end{displaymath}
is

\begin{itemize}%
\item a lift in

\begin{displaymath}
\itexarray{
    && Y_n
    \\
    & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow
    \\
    B_n &\to& X_n
  }
\end{displaymath}

\item such that it makes

\begin{displaymath}
\itexarray{
     B_n &\stackrel{\gamma_n}{\to}& Y_n
     \\
     \downarrow && \downarrow
     \\
     Match_n(B) &\to& Match_n(Y)
  }
\end{displaymath}
commute.



\end{itemize}
By lemma \ref{SkeletalExtension}, this is precisely the data that characterizes an extension of $\gamma_{\leq (n-1)}$ to $\gamma_{\leq n}$.

By assumption, the left vertical morphism in \eqref{MainLiftingDiagram} is a cofibration in $[G_n, \mathcal{C}]_{proj/inj}$, and the right vertical morphism is a fibration there. Therefore to get the lift and hence complete the induction step, it is now sufficient to show that the left morphism is also a weak equivalence, hence is a weak equivalence in $\mathcal{C}$ over each $s \in S$.

Also by assumption we have that $Latch_n(f)_s$ is an acyclic cofibration in $\mathcal{C}$ for all $s$. Hence so is its pushout $A_s \to 
(A_s \coprod_{Latch_n(A)_s} Latch_n(B)_s)$. The morphism $v_n(s)$ finally sits in the diagram

\begin{displaymath}
\itexarray{
    Latch_n(A)_s
    &\to&
    A_s &\underoverset{\simeq}{f_s}{\to}& B_s
    \\
    \downarrow^{\mathrlap{\simeq}} &&
    \downarrow^{\mathrlap{\simeq}}  & \nearrow_{\mathrlap{v_n(s)}}
    \\
    Latch_n(B)_s &\to& (A_s \coprod_{Latch_n(A)_s} Latch_n(B)_s)
  }
\end{displaymath}
and so is a weak equivalence by [[two-out-of-three]].

\end{proof}
\begin{lemma}
\label{LatchingIntertwiner}\hypertarget{LatchingIntertwiner}{}
Suppose $\phi : R \to S$ is a morphism of [[generalized Reedy categories]] such that the induced square (see prop. \ref{DiagramOfRestrictions})

\begin{displaymath}
\itexarray{
    R^+((k)) &\stackrel{cod_k^R}{\to}& G_k(R)
    \\
    \downarrow^{\mathrlap{\phi_k^+}} && \downarrow^{\mathrlap{\phi_k}}
    \\
    S^+((k)) &\stackrel{cod_k^S}{\to}& G_k(S)
  }
\end{displaymath}
is a pullback (in the 1-category [[Cat]]). Then for each $X \in [S, \mathcal{C}]$, there is a [[natural isomorphism]]

\begin{displaymath}
Latch_k(\phi^*(X)) \stackrel{\simeq}{\to} \phi_k^*(Latch_k X)
  \,,
\end{displaymath}
Given a Reedy category $S$, examples of $\phi$ satisfying this condition include

\begin{enumerate}%
\item $\phi := dom_n \circ i_n : S^+(n) \to S$;


\item $\phi := : S_{\leq n} \hookrightarrow S$.



\end{enumerate}
\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 4.4}).

\begin{proof}
The pullback square is part of the diagram

\begin{displaymath}
\itexarray{
    R
     &\stackrel{dom_k^R}{\leftarrow}&
    R^+((k)) &\stackrel{cod_k^R}{\to}& G_k(R)
    \\
    \downarrow^\phi
     &&
    \downarrow^{\mathrlap{\phi_k^+}} && \downarrow^{\mathrlap{\phi_k}}
    \\
    S
     &\stackrel{dom_k^S}{\leftarrow}&
    S^+((k)) &\stackrel{cod_k^S}{\to}& G_k(S)
  }
\end{displaymath}
whose rows define, by prop. \ref{DiagramOfRestrictions}, the latching objects by pull-push. Since the pullback square, being the pullback of an [[opfibration]] (the [[codomain opfibration]]), satisfies the [[Beck-Chevalley condition]] (by the fact discussed \href{http://ncatlab.org/nlab/show/Beck-Chevalley+condition#PullbacksOfOpfibrations}{here}), we find the intertwining isomorphism as follows:

\begin{displaymath}
\begin{aligned}
     Latch_k \phi^* X 
     & \simeq 
     (cod^R_k)_! (dom^R_k)^* \phi^* X
     \\
      & \simeq 
      (cod^R_k)_! (\phi^+_k)^* (dom^S_k)^* X
     \\
     & \simeq
     (\phi_k)^* (cod^R_k)_! (dom^S_k)^* X
     \\
     & \simeq
     (\phi_k)^*
     Latch_k X
  \end{aligned}
  \,.
\end{displaymath}
Now concerning the two examples.

By definition we have

\begin{displaymath}
(S^+(n))^+((k))
 \simeq
  \left\{
  \itexarray{  
    t &\stackrel{\in S^+}{\to}& t'
    \\
    \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} && non\; invertible
    \\
    s &\stackrel{\simeq}{\to}& s' && deg = k
    \\
    \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} && non \; invertible
    \\
    r &\stackrel{id}{\to}& r  && deg = n
  }
  \right\}
  \,,
\end{displaymath}
where on the right the vertical sequences in $S$ indicate objects in $(S^+(m))^+((k))$ and the whole diagram on the right indicates a morphism there.

One sees that this is indeed the fiber product as claimed.

\end{proof}
We now show that the extra condition in prop. \ref{LiftingLemma} is in fact automatic.

\begin{lemma}
\label{ExtraPropertyOfAcyclicCofibrations}\hypertarget{ExtraPropertyOfAcyclicCofibrations}{}
Let $f : X \to Y$ in $[S, \mathcal{C}]$ be a Reedy cofibration, which is a weak equivalence on all objects of degree $\lt n$. Then the morphism $Latch_n(f) :  Latch_n(X) \to  Latch_n(Y)$ is over each $s \in S$ an acyclic cofibration in $\mathcal{C}$

\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 5.3}).

\begin{proof}
We show this by induction over $n$, using the skeletal filtration def. \ref{SkeletaByAdjunction}. For $n = 0$ we have for all $X$ that $Latch_n X = sk_{-1} X = \emptyset$, and hence the condition is trivially satisfied.

So assume now that the statement has been shown for all $(k \lt n)$, then we need to show that $i_n^* Latch_n f$ is an acyclic cofibration in $[Obj(S)_n, \mathcal{C}]$, hence that every square of the form

\begin{displaymath}
\itexarray{   
    i_n^* Latch_n A &\to& Y
    \\
    \downarrow^{\mathrlap{i_n^* Latch_n f}} && \downarrow^{g}
    \\
    i_n^* Latch_n B &\to& X
  }
\end{displaymath}
with $g$ a fibration in $[Obj(S)_n, \mathcal{C}]_{proj/inh}$ (hence over every object of degree $n$) has a lift. Since by lemma \ref{DiagramOfRestrictions} we have

\begin{displaymath}
i^*_n Latch_n := i_n^* cod_! dom_n^* \simeq cod_! i_n^* dom_n^*
\end{displaymath}
such a filler is equivalently a filler in

\begin{equation}
\itexarray{
     i_n^* dom_n^* A &\to& cod_n^* Y
     \\
     {}^{\mathllap{i_n^* dom_n^*}}{}\downarrow && \downarrow
     \\
     i_n^* dom_n^* B &\to& cod_n^* X
  }
\label{SomeLiftingDiagram}\end{equation}
being a diagram in $[S^+(n), \mathcal{C}]$.

We now establish such a filler by using lemma \ref{LiftingLemma} with the Reedy category in question being now $S^+(n)$. This has only $+$-morphisms and hence Reedy fibrations here are objectwise fibrations, so the morphism on the right is a Reedy fibration over $S^+(n)$.

Moreover, $i_n^* dom_n^* f$ is a Reedy weak equivalence in this structure, since all its objects have degree $\lt n$. It is now sufficient to show that the assumptions of lemma \ref{LiftingLemma} are satisfied over $S^+(n)$, to obtain the lift.

By lemma \ref{LatchingIntertwiner} the functor

\begin{displaymath}
(dom_n i_n)^* : [G_k(S), \mathcal{C}] \to [G_k(S^+(n)), \mathcal{C}]
\end{displaymath}
intertwines the latching objects on both sides. Therefore we have an isomorphism between the relative latching morphism of interest

\begin{displaymath}
(dom_n i_n)^* A \coprod_{Latch_k (dom_n i_n)^* A} Latch_k (dom_n i_n)^* k \to (dom_n i_n)^* B
\end{displaymath}
and the morphism

\begin{displaymath}
(dom_n i_n)_k^*\left(
    A_k \coprod_{Latch_k(A)} Latch_k(B)
    \to
    B_k
  \right)
  \,.
\end{displaymath}
Since $(dom_n i_n)_k$ is a [[faithful functor]] between groupoids, $(dom_n i_n)_k^*$ preserves cofibrations in the projective structure (and trivially does so always in the injective structure), and so the above relative latching morphism is a cofibration, hence $(dom_n i_n)^*(f)$ is a Reedy cofibration. Similarly, since $Latch_k(f)$ is an acyclic cofibration by induction hypothesis, so is $Latch_k((dom_n i_n)_k^* f)$. This way the assumption of lemma \ref{LiftingLemma} are checked for \eqref{SomeLiftingDiagram} and so we do have a lift.

\end{proof}
\begin{lemma}
\label{ExtraPropertyOfAcyclicFibrations}\hypertarget{ExtraPropertyOfAcyclicFibrations}{}
Dually, let $f : X \to Y$ in $[S, \mathcal{C}]$ be a Reedy fibration which is a weak equivalence over all objects of degree $\lt n$. Then the morphism $Match_n(f) :  Match_n(X) \to  Match_n(Y)$ is over each $s \in S$ an acyclic fibration in $\mathcal{C}$.

\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 5.4}).

\begin{proof}
(Essentially the dual proof to that above. Except for one slight difference in the last part. Here -- and only here -- do we need the last clause in the definition of [[generalized Reedy category]], the one that says that isomorphisms see the maps in $S_-$ as epimorphisms.)

\end{proof}
We can finally conclude:

\begin{prop}
\label{}\hypertarget{}{}
In the generalized Reedy model structure, def. \ref{ReedyModelStructure},

\begin{itemize}%
\item acyclic cofibrations have the [[left lifting property]] against fibrations;


\item cofibrations have the [[left lifting property]] against acyclic fibrations.



\end{itemize}
\end{prop}
\begin{proof}
By lemma \ref{ExtraPropertyOfAcyclicCofibrations} every acyclic Reedy cofibration induces a weak equivalence under $Latch_n$. By lemma \ref{LiftingLemma} this implies the left lifting property against Reedy fibrations. Dually for the second statement.

\end{proof}
\hypertarget{Factorization}{}\subsubsection*{{Factorization}}\label{Factorization}

We demonstrate the factorization axiom in the Reedy model structure, def. \ref{ReedyModelStructure}.

\begin{lemma}
\label{AcyclicCoFibrationsByLatchingMatching}\hypertarget{AcyclicCoFibrationsByLatchingMatching}{}
A morphism $f : X \to Y$ in $[S, \mathcal{C}]$ is both a cofibration and a weak equivalence according to def. \ref{ReedyModelStructure} precisely if the $n$th relative latching morphism (def. \ref{GlobalLatching})

\begin{displaymath}
X_n \coprod_{Latch_n(X)} Latch_n(Y) \to Y_n
\end{displaymath}
is an acyclic cofibration in $[G_n(S), \mathcal{C}]_{proj/inj}$ for all $n \in \mathbb{N}$.

Dually, a morphism $f : X \to Y$ in $[S, \mathcal{C}]$ is both a fibration and a weak equivalence according to def. \ref{ReedyModelStructure} precisely if the $n$th relative matching morphism (def. \ref{GlobalLatching})

\begin{displaymath}
X_n \to Y_n \prod_{Match_n(X)} Match_n(Y)
\end{displaymath}
is an acyclic fibration in $[G_n(S), \mathcal{C}]_{proj/inj}$ for all $n \in \mathbb{N}$.

\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, prop. 5.6}).

\begin{proof}
By definition, the morphisms $f_n : X_n \to Y_n$ factor as

\begin{displaymath}
f_n : 
  X_n 
   \stackrel{u_n}{\to}
  X_n
  \coprod_{Latch_n(X)} Latch_n (Y)
   \stackrel{v_n}{\to}
  Y_n 
  \,.
\end{displaymath}
If now $f$ is an acyclic Reedy cofibration, then by lemma \ref{ExtraPropertyOfAcyclicCofibrations} the morphism $Latch_n(X) \to Latch_n(Y)$ is over each object an acyclic cofibration in $\mathcal{C}$ and then so is $u_n$ above, being the pushout of this morphism. It follows by [[two-out-of-three]] that also $v_n$ is a weak equivalence for all $n$.

Conversely, assume that all $v_n$ here are weak equivalences. We show by induction on $n$ that then also the $u_n$ are weak equivalences, and hence that $f$ is a Reedy weak equivalence.

For $n = 0$ we have $u_0 = id$, and so this case is satisfied. So assume now that all $u_k$ for $k \lt n$ are weak equivalences. Then the assumptions of lemma \ref{ExtraPropertyOfAcyclicCofibrations} are again satisfied, and it follows that $Latch_n(f) : Latch_n(X) \to Latch_n(Y)$ is over each object an acyclic cofibration. Accordingly, so is $u_n$, being its pushout. Therefore, by induction, all $u_n$ are, in particular, weak equivalences.

The argument for fibrations is dual to this.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Every morphism in $[S, \mathcal{C}]$ factors as an acyclic Reedy cofibration (according to def. \ref{ReedyModelStructure}) followed by a Reedy fibration, and it factors as a Reedy cofibration followed by an acyclic Reedy fibration

\end{prop}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, page 18}).

\begin{proof}
Let $f : X \to Y$ be any morphism in $[S, \mathcal{C}]$. We construct a factorization into an acyclic cofibration followed by a fibration by induction on the degree, i.e. by induction over the restrictions along $t_n^* : [S, \mathcal{C}] \to [S_{\leq n}, \mathcal{C}]$. The other case (cofibration followed by acyclic fibration) works dually.

For $n = 0$ we have $S_{\leq 0} = G_0(S)$ and we factor $f_0$ in the model structure $[G_0(S), \mathcal{C}]_{proj/inj}$

\begin{displaymath}
f_0 : X_0 \stackrel{\simeq}{\to} A_0 \to Y_0
  \,.
\end{displaymath}
Now assume for some $n \in \mathbb{N}$ that a factorization of

\begin{displaymath}
f_{\leq (n-1)}
 : 
  X_{\leq (n-1)}
  \stackrel{\simeq}{\to}
  A_{\leq (n-1)}
  \to
  Y_{\leq (n-1)}
\end{displaymath}
has been found. This induces the commutative diagram

\begin{displaymath}
\itexarray{
    Latch_n(X) &\to & Latch_n(A) &\to& Latch_n(Y)
    \\
    \downarrow && && \downarrow
    \\
    X_n && && Y_n
    \\
    \downarrow && && \downarrow
    \\
    Match_n(X) &\to & Match_n(A) &\to& Match_n(Y) 
  }
  \,.
\end{displaymath}
This diagram in turn induces a morphism

\begin{displaymath}
X_n \coprod_{Latch_n(X)} Latch_n(A)
  \to 
  Y_n \prod_{Match_n(Y)} Match_n(A)
\end{displaymath}
in $[G_n(S), \mathcal{C}]$, which we may factor as a trivial cofibration followed by a fibration

\begin{displaymath}
X_n \coprod_{Latch_n(X)} Latch_n(A)
   \stackrel{\simeq}{\to}
   A_n
  \to 
  Y_n \prod_{Match_n(Y)} Match_n(A)
\end{displaymath}
in $[G_n(S), \mathcal{C}]_{proj/inj}$.

By lemma \ref{SkeletalExtension}, the ``righmost component'' of this data defines an extension of $A_{\leq (n-1)}$ to $A_{\leq n}$. The ``leftmost component'' defines a factorization of $f_n$ and the ``middle component'' says that this consistently extends the previously obtained factorization of $f_{\leq n}$. With this, finally the two morphisms say that this new factorization is by an acyclic Reedy cofibration (by lemma \ref{AcyclicCoFibrationsByLatchingMatching}) followed by a Reedy fibration (by definition).

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

\hypertarget{HomotopySkeletalFiltration}{}\subsubsection*{{Homotopy skeletal filtration and coskeleton tower}}\label{HomotopySkeletalFiltration}

We discuss conditions on an object $X$ such that the skeletal/coskeletal towers discussed \hyperlink{SkeletaAndCoskeleta}{above} are ``homotopy good'' in that they exhibit $X$ not just as the colimit/limit over the tower, but as the [[homotopy colimit]]/[[homotopy limit]].

\begin{lemma}
\label{QuillenPropertyOfCoSkeleta}\hypertarget{QuillenPropertyOfCoSkeleta}{}
For all $n \in \mathbb{N}$

\begin{enumerate}%
\item The left Kan extension

\begin{displaymath}
(t_n)_! : [S_{\leq n}, \mathcal{C}]_{gReedy} \to [S,\mathcal{C}]_{gReedy}
\end{displaymath}
is a [[left Quillen functor]].


\item The right Kan extension

\begin{displaymath}
(t_n)_* : [S_{\leq n}, \mathcal{C}]_{gReedy} \to [S,\mathcal{C}]_{gReedy}
\end{displaymath}
is a [[right Quillen functor]].


\item The restriction functor

\begin{displaymath}
(t_n)^* : [S, \mathcal{C}]_{gReedy} \to [S_{\leq n},\mathcal{C}]_{gReedy}
\end{displaymath}
is both a left and a right Quillen functor.



\end{enumerate}
\end{lemma}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, lemma 6.4}).

\begin{prop}
\label{ReedyCoFibrantImpliesCoFibrantTowers}\hypertarget{ReedyCoFibrantImpliesCoFibrantTowers}{}
\begin{enumerate}%
\item If $X \in [S, \mathcal{C}]$ is Reedy cofibrant according to def. \ref{ReedyModelStructure}, then all $sk_n X$ are Reedy cofibrant and the canonical morphisms $sk_k X \to sk_l X$ are Reedy cofibrations.


\item If $X \in \mathcal{C}$ is Reedy fibrant according to def. \ref{ReedyModelStructure} then all $cosk_n X$ are Reedy fibrant and the canonical morphisms $cosk_l X \to cosk_k X$ are Reedy fibrations.



\end{enumerate}
\end{prop}
This is (\hyperlink{BergerMoerdijk}{Ber-Moer, prop. 6.5}).

\begin{proof}
We discuss the skeleta. The case of coskeleta is dual.

By lemma \ref{IdempotenceOfCoSkeleta} it is sufficient to consider the case $sk_n X \to sk_\infty X = X$.

To check that this is a Reedy cofibration if $X$ is Reedy cofibrant, consider the diagram that induces the relative latching object for this morphism

\begin{displaymath}
\itexarray{
    Latch_k(sk_n X) &\to& Latch_k (X)
    \\
    \downarrow && \downarrow
    \\
    sk_n(X)_k &\to& X_k
  }
  \,.
\end{displaymath}
For $k \leq n$, the horizontal morphisms are both isomorphisms. Because by lemma \ref{LatchingIsSkeleton} the top morphism is

\begin{displaymath}
(sk_{k-1} sk_n(X))_k \to (sk_{k-1} X)_k
\end{displaymath}
and this is an iso by lemma \ref{IdempotenceOfCoSkeleta}. The bottom morphism is isomorphic to $(t_k^* sk_{n}(X))_k \to (t_k^* X)_k$ which is isomorphic to the identity by the kind of argument in lemma \ref{IdempotenceOfCoSkeleta}.

Therefore also the relative latching morphism is an isomorphism in this case (use that pushouts of isos are isos and use 2-out-of-3 for isos), hence in particular a cofibration.

Similarly, for $k \gt n$ the left vertical map is an isomorphism, so that the relative latching morphism in this case is $Latch_k(X) \to X_k$, which is a cofibration by the assumption that $X$ is Reedy cofibrant.

Finally, that $sk_n X$ is cofibrant follows directly from lemma \ref{QuillenPropertyOfCoSkeleta}.

\end{proof}
\hypertarget{relation_to_other_model_structures}{}\subsubsection*{{Relation to other model structures}}\label{relation_to_other_model_structures}

(\ldots{})

\hypertarget{example}{}\subsection*{{Example}}\label{example}

\begin{itemize}%
\item Every ordinary [[Reedy category]] is a generalized Reedy category, and in this case the above model structure reduces to the traditional [[Reedy model structure]].


\item The [[model structure for dendroidal complete Segal spaces]] is a [[Bousfield localization of model categories|left Bousfield localization]] of the generalized Reedy model structure over the [[tree category]].



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

\begin{itemize}%
\item [[Joyal-Tierney calculus]]

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

\begin{itemize}%
\item [[Clemens Berger]] and [[Ieke Moerdijk]], \emph{On an extension of the notion of Reedy category}, Math. Z. (2008) (\href{http://arxiv.org/abs/0809.3341}{arXiv:0809.3341})

\end{itemize}


\end{document}