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\begin{document}

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\section*{generalized Reedy category}

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

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

[[!include model category theory - contents]]

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

[[!include homotopy - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{generalized_reedy_categories}{Generalized Reedy categories}\dotfill \pageref*{generalized_reedy_categories} \linebreak
\noindent\hyperlink{prima_facie_comparison_between_cisinski_and_bergermoerdijk}{Prima Facie comparison between Cisinski and Berger-Moerdijk.}\dotfill \pageref*{prima_facie_comparison_between_cisinski_and_bergermoerdijk} \linebreak
\noindent\hyperlink{presheaves_on_reedy_categories}{Presheaves on Reedy categories.}\dotfill \pageref*{presheaves_on_reedy_categories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{NormalMorphisms}{Normal monomorphisms (Cisinski)}\dotfill \pageref*{NormalMorphisms} \linebreak
\noindent\hyperlink{model_category_structure_on_presheaves}{Model category structure on presheaves}\dotfill \pageref*{model_category_structure_on_presheaves} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{bergermoerdijk_type}{Berger-Moerdijk type}\dotfill \pageref*{bergermoerdijk_type} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{the_class_of_crossed_group_sites}{The class of crossed group sites}\dotfill \pageref*{the_class_of_crossed_group_sites} \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 notion of \emph{[[Reedy category]]}, though useful, is in some contexts inconveniently restrictive, since no Reedy category can contain any nonidentity [[isomorphisms]]. This is problematic for many ``[[geometric shape for higher structures|shape]] categories'' such as Connes' [[category of cycles]] $\Lambda$, [[Segal's category Gamma|Segal's category]] $\Gamma$, the [[tree category]] $\Omega$, and so on. The notion of \emph{generalized Reedy category} lifts this restriction, while maintaining the truth of the main theorem about Reedy categories: the existence of the [[Reedy model structure]].

In fact, there are two notions of generalized Reedy category in the literature. Cisinski's ``cat\'e{}gories squelletiques'' (Ch. 8 in \hyperlink{Cisinski}{\emph{PCMH}}) provide a natural generalization of the [[simplex category]], so that diagrams based on them behave much like [[simplicial objects]]. They were introduced primarily for the purposes of modeling [[homotopy types]]. The \hyperlink{BergerMoerdijk}{``generalized Reedy categories''} of Berger and Moerdijk are a strictly broader generalization suitable e.g. for describing [[dendroidal sets]]. They were introduced for the purposes of modelling more general classes of structure, particularly [[operads]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{generalized_reedy_categories}{}\subsubsection*{{Generalized Reedy categories}}\label{generalized_reedy_categories}

\begin{defn}
\label{BMGeneralizedReedyCategory}\hypertarget{BMGeneralizedReedyCategory}{}
A \textbf{Berger-Moerdijk generalized Reedy category} is a [[category]] $R$ together with two [[wide subcategories]] $R_+$ and $R_-$, and a [[function]] $d\colon ob(R)\to \alpha$ called \emph{degree}, for some [[ordinal]] $\alpha$, such that

\begin{enumerate}%
\item every non-isomorphism in $R_+$ raises degree,


\item every non-isomorphism in $R_-$ lowers degree,


\item every isomorphism in $R$ preserves degree,


\item $R_+\cap R_-$ is the [[core]] of $R$ (equivalently, every isomorphism is in both $R_+$ and $R_-$, i.e. they are not just wide but [[pseudomonic functor|pseudomonic]] subcategories),


\item every morphism $f$ factors as a map in $R_-$ followed by a map in $R_+$, uniquely up to isomorphism,


\item if $f\in R_-$ and $\theta$ is an isomorphism such that $\theta f = f$, then $\theta = 1$ (isomorphisms see the maps in $R_-$ as [[epimorphism|epis]]).



\end{enumerate}
The last condition implies that the isomorphism in the penultimate condition must be unique. It is not self-dual, but has an obvious dual version. A BM generalized Reedy category is said to be \textbf{dualizable} if it satisfies both this condition and its dual.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A [[morphism]] of generalized Reedy category $S \to R$ is a [[functor]] whish takes $S_+$ to $R_+$, takes $S_-$ to $R_-$ and preserves the degree.

\end{defn}
This appears as (\hyperlink{BergerMoerdijk}{Berger-Moerdijk, def. 1.1}).

\begin{remark}
\label{}\hypertarget{}{}
Generalized Reedy category structures (as opposed to ordinary structures!) can always be transported along [[equivalence of categories]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For a \textbf{Cisinski generalized Reedy category}, the final condition in def. \ref{BMGeneralizedReedyCategory} is replaced by

\begin{itemize}%
\item every morphism in $R_-$ admits a [[section]], and two parallel morphisms in $R_-$ are equal precisely when they have the same sections.

\end{itemize}
\end{defn}
For clarity, in the context of generalized Reedy categories, an ordinary [[Reedy category]] may be called a \emph{strict Reedy category}.

\hypertarget{prima_facie_comparison_between_cisinski_and_bergermoerdijk}{}\subsubsection*{{Prima Facie comparison between Cisinski and Berger-Moerdijk.}}\label{prima_facie_comparison_between_cisinski_and_bergermoerdijk}

The only difference between the Cisinski notion and the Berger-Moerdijk notion is in the final condition -- let's say, between the \emph{Cisinski condition} and the \emph{Berger-Moerdijk condition}.

It's easy to see that the Cisinski condition is \textbf{strictly stronger} than the Berger-Moerdijk condition. The Berger-Moerdijk condition asks that $R_-$ arrows be something less than epimorphic in $R$. By comparison, the Cisinski condition asks that $R_-$ arrows be \emph{actually} epimorphic in $R$, in fact that they be \emph{split} epimorphic, and more.

\hypertarget{presheaves_on_reedy_categories}{}\subsubsection*{{Presheaves on Reedy categories.}}\label{presheaves_on_reedy_categories}

For $R$ a generalized Reedy category, and $X$ a [[presheaf]] on $R$, there are the evident analogue notions of $n$-cells in $X$, degenerate $n$-cells, faces, boundaries, horns, etc.

(..)

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

\hypertarget{NormalMorphisms}{}\subsubsection*{{Normal monomorphisms (Cisinski)}}\label{NormalMorphisms}

Let $A$ be a Cisinski generalized Reedy category.

\begin{defn}
\label{}\hypertarget{}{}
An object $a \in A$ is called \emph{degenerate} precisely if there is a non-isomorphism out of $a$ in $A_-$.

\end{defn}
See (\hyperlink{Cisinski}{Cisinski, prop. 8.1.9}).

\begin{prop}
\label{}\hypertarget{}{}
For every object $a \in A$ there exists a morphism $\pi : a \to b$ in $A_-$ with $b$ non-degenerate.

\end{prop}
This is (\hyperlink{Cisisnki}{Cisinski, prop. 8.1.13}).

Let $X$ be a [[presheaf]] over $A$.

\begin{defn}
\label{}\hypertarget{}{}
For $a \in A$, a cell $v \in X(a)$ is called \textbf{degenerate} precisely if there is a morphism $\alpha : a \to a'$ in $A_-$ and a cell $u \in$

\begin{displaymath}
X(\alpha) : u \mapsto v
\end{displaymath}
\end{defn}
See (\hyperlink{Cisinski}{Cisinski, cor. 8.1.10}).

Write $A/X$ for the [[category of elements]] of $X$.

\begin{defn}
\label{}\hypertarget{}{}
For $a \in A$ an $a$-cell $v \in X(a)$ is called \textbf{dominant} if $(a,v) \in A/X$ has trivial [[automorphism group]].

An $a$-cell $u \in X(a)$ is called \textbf{normal} if there is a morphism $\alpha : a \to b$ in $A_-$ with $b$ non-degenerate, and a dominant $b$-cell $v \in X(b)$, such that

\begin{displaymath}
X(\alpha) : v \mapsto u
  \,.
\end{displaymath}
The presheaf $X$ is called \textbf{normal} if all its cells are normal.

\end{defn}
See (\hyperlink{Cisinski}{Cisinski, 8.1.23}).

\begin{remark}
\label{}\hypertarget{}{}
A non-degenerate cell is normal precisely if it is dominant.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
$X$ is normal precisely if all its non-degenerate cells are dominant.

\end{prop}
This is (\hyperlink{Cisinski}{Cisinski, cor. 8.1.25}).

Let $f : X \to Y$ be a morphism of [[presheaves]] over $A$.

\begin{defn}
\label{}\hypertarget{}{}
The morphism $f : X \to Y$ is called \textbf{normal} if every cell of $Y$ not in the [[image]] of $f$ is dominant.

\end{defn}
This is (\hyperlink{Cisinski}{Cisinski, 8.1.30}).

\begin{example}
\label{}\hypertarget{}{}
Every [[monomorphism]] between normal presheaves is normal.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
The class of normal monomorphisms in $PSh(A)$ is closed under

\begin{itemize}%
\item [[pushouts]];


\item [[transfinite composition]];


\item [[retracts]].



\end{itemize}
In fact, the class of normal monomorphisms is that generated under these operations from the boundary inclusions $I := \{\partial a \hookrightarrow a\}$.

\end{prop}
This is (\hyperlink{Cisinski}{Cisinski, prop. 8.1.31, 8.1.35}).

\hypertarget{model_category_structure_on_presheaves}{}\subsubsection*{{Model category structure on presheaves}}\label{model_category_structure_on_presheaves}

(\ldots{}) [[generalized Reedy model structure]]

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

\hypertarget{bergermoerdijk_type}{}\subsubsection*{{Berger-Moerdijk type}}\label{bergermoerdijk_type}

\hypertarget{specific_examples}{}\paragraph*{{Specific examples}}\label{specific_examples}

\begin{itemize}%
\item Any [[Reedy category]] is a generalized Reedy category, in particular the [[simplex category]].


\item Any (finite) [[groupoid]] $G$ is also a generalized Reedy category, with $G_+ = G_- = G$.


\item Connes' [[category of cycles]] $\Lambda$.


\item [[Segal's category Gamma|Segal's category]] $\Gamma$.


\item The Moerdijk-Weiss [[tree category]] $\Omega$ is generalized Reedy. The degree is given by the number of vertices in a tree.

See also the discussion at \emph{[[dendroidal set]]} and \emph{[[model structure on dendroidal sets]]}.


\item Any generalized [[direct category]] or generalized [[inverse category]] is also a generalized Reedy category, in which either $R_-$ or $R_+$ consists only of the isomorphisms.



\end{itemize}
\hypertarget{the_class_of_crossed_group_sites}{}\paragraph*{{The class of crossed group sites}}\label{the_class_of_crossed_group_sites}

\begin{defn}
\label{}\hypertarget{}{}
For $S$ a [[small category]], a \textbf{crossed $S$-group} is a [[presheaf]] $G : S^{op} \to Set$ equipped with

\begin{enumerate}%
\item for each object $s \in S$ a group structure on $G_s$;


\item for all $s, r\in S$ a left $G_r$-[[action]] on the [[hom-set]] $S(s,r)$ ;



\end{enumerate}
such that for all morphisms $\alpha : s \to r$ and $\beta : t \to s$ in $S$ and $g,h \in G_r$ we have

\begin{enumerate}%
\item $g_*( \alpha \circ \beta)  = g_*(\alpha) \circ \alpha^*(g)_* \beta$;


\item $g_* (id_r) = id_r$;


\item $\alpha^* (g \cdot h) = h_*(\alpha)^*(g)\cdot \alpha^*(h)$;


\item $\alpha^*(e_r) = e_s$;



\end{enumerate}
where $g_*$, $h_*$ denotes the group action and $\alpha^* : G_r \to G_s$ the presheaf map.

The \textbf{total category} $S G$ of an crossed $S$-group $G$ is the category with the same objects as $S$, and with morphisms $r \to s$ being pairs $(\alpha, g) \in S(s,r)\times G_r$ and with composition defined by

\begin{displaymath}
(\alpha, g) \circ (\beta, h)
  = 
  (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BergerMoerdijk}{Ber-Moe, def. 2.1}).

\begin{defn}
\label{}\hypertarget{}{}
If $S$ is equipped with a generalized Reedy structure, then an $S$-crossed group $G$ is called \textbf{compatible} with that generalized Reedy structure if

\begin{enumerate}%
\item the $G$-action respects $S^+$ and $S^-$;


\item if $\alpha : r \to s$ is in $S^-$ and $g \in G_s$ such that $\alpha^* (g) = e_r$ and $g_*(\alpha) = \alpha$, then $g = e_s$.



\end{enumerate}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The category $\Omega_{pl}$ of planar finite rooted [[trees]] is a strict [[Reedy category]]. The category $\Omega$ of non-planar finite rooted [[trees]] is the total category of an $\Omega_{pl}$-crossed group which to a planar tree $T$ assigns its group of non-planar automorphisms.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Let $S$ be a strict [[Reedy category]] and let $G$ be a compatible $S$-crossed group. Then there exists a unique dualizabe generalized Reedy structure on $S G$ for which the embedding $S \hookrightarrow S G$ is a morphism of generalized Reedy categories.

\end{prop}
(\hyperlink{BergerMoerdijk}{Ber-Moe, prop. 2.10}).

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

\begin{itemize}%
\item A ``more generalized'' notion is a [[c-Reedy category]].

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

Cisinski's notion of generalized Reedy category appears as def 8.1.1 in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{[[joyalscatlab:Les préfaisceaux comme type d'homotopie|Les préfaisceaux comme modèles des types d'homotopie]]}, Ast\'e{}risque, Volume 308, Soc. Math. France (2006), 392 pages (\href{http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf}{pdf})

\end{itemize}
The Berger-Moerdijk definition of generalized Reedy category appears in

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

\end{itemize}
[[!redirects generalized Reedy categories]] [[!redirects Berger-Moerdijk generalized Reedy category]] [[!redirects Berger-Moerdijk generalized Reedy categories]] [[!redirects Cisinski generalized Reedy category]] [[!redirects Cisinski generalized Reedy categories]]



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