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\begin{document}

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{degeneracies_are_split}{Degeneracies are split}\dotfill \pageref*{degeneracies_are_split} \linebreak
\noindent\hyperlink{degeneracies_in_subobjects}{Degeneracies in subobjects}\dotfill \pageref*{degeneracies_in_subobjects} \linebreak
\noindent\hyperlink{all_presheaves_are_reedy_monomorphic}{All presheaves are ``Reedy monomorphic''}\dotfill \pageref*{all_presheaves_are_reedy_monomorphic} \linebreak
\noindent\hyperlink{reedy__injective}{Reedy = injective}\dotfill \pageref*{reedy__injective} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{elegant Reedy category} is a [[Reedy category]] $R$ such that the following \emph{equivalent} conditions hold

\begin{itemize}%
\item For every [[monomorphism]] $A\hookrightarrow B$ of [[presheaves]] on $R$ and every $x\in R$, the induced map $A_x \amalg_{L_x A} L_x B \to B_x$ is a monomorphism.


\item Every [[span]] of codegeneracy maps in $R_-$ has an [[absolute colimit|absolute]] [[pushout]] in $R_-$.


\item Every element of a presheaf $R$ is a degeneracy of some nondegenerate element in a unique way.



\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The principal theorem about elegant Reedy categories is that the [[Reedy model structure]] on [[presheaves]] (i.e. contravariant diagrams) over an elegant Reedy category coincides with the [[injective model structure]]. This is not true for presheaves valued in \emph{any} model category, only well-behaved ones. We clarify the necessary conditions by building up to this theorem in stages, adding hypotheses on the codomain of the presheaves as necessary.

\hypertarget{degeneracies_are_split}{}\subsubsection*{{Degeneracies are split}}\label{degeneracies_are_split}

\begin{lemma}
\label{DegSplit}\hypertarget{DegSplit}{}
If $R$ is elegant, then every codegeneracy map (i.e. morphism in $R_-$) is a [[split epimorphism]].

\end{lemma}
\begin{proof}
Let $f:x\to y$ be a codegeneracy map; then the span $y \xleftarrow{f} x \xrightarrow{f} y$ has an absolute pushout, consisting of say $g:y\to z$ and $h:y\to z$ with $g f = h f$. This absolute pushout is preserved by $R(z,-)$, so $1_z\in R(z,z)$ must be the image under $g$ or $h$ of some map $s:z\to y$; WLOG say it is $g$, so we have $1_z = g s$. Now we have $s h:y\to y$ and $1_y$ such that $g s h = h = h 1_y$, and our absolute pushout is preserved by $R(y,-)$, so there must be a zigzag of elements in $R(y,z)$ relating $s h$ to $1_y$. At one end of that zigzag, there must be a $t:y\to x$ such that $f t = 1_y$; hence $f$ is split epi.

\end{proof}
\begin{lemma}
\label{MonoPreservesDeg}\hypertarget{MonoPreservesDeg}{}
For every monomorphism $A\hookrightarrow B$ of presheaves on $R$, every nondegenerate element of $A$ remains nondegenerate in $B$.

\end{lemma}
\begin{proof}
Let $a$ be a nondegenerate element of $A_x$, for some $x\in R$, $f : x \to y$ a codegeneracy map, and $b\in B_y$ such that $B_f b = a$. We have to show that $f = \id$. By Lemma \ref{DegSplit}, $f$ has a section $s: y \to x$, hence $B_s a = B_s B_f b = b$, which implies that $b \in A_y$. Since $a$ is nondegenerate, it follows that $f = \id$.

\end{proof}
\hypertarget{degeneracies_in_subobjects}{}\subsubsection*{{Degeneracies in subobjects}}\label{degeneracies_in_subobjects}

\begin{lemma}
\label{DegSub}\hypertarget{DegSub}{}
Let $R$ be elegant and $f:x\to y$ a codegeneracy in $R$. Let $M$ be any category, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$. Then the following naturality square is a pullback:

\begin{displaymath}
\itexarray{ A_y & \xrightarrow{A_f} & A_x \\
  {}^{\mu_y}\downarrow & & \downarrow^{\mu_x} \\
  B_y & \xrightarrow{B_f} & B_x  }
\end{displaymath}
\end{lemma}
\begin{proof}
This depends only on the fact that $f$ is split epi in $R$. Let $s:y\to x$ be a section of it, and let $P$ be the pullback of $B_f$ and $\mu_x$, with projections $p:P\to A_x$ and $q:P\to B_y$ with $\mu_x p = B_f q$, and an induced map $\phi:A_y \to P$ such that $p \phi = A_f$ and $q\phi = \mu_y$.

We claim that $A_s p : P \to A_y$ is an inverse of $\phi$, making it an isomorphism. On the one hand we have $A_s p \phi = A_s A_f = A_{f s} = 1$. On the other, to show that $\phi A_s p = 1$ it suffices to show that $p \phi A_s p = p$ and $q \phi A_s p = q$. For the first, since $\mu_x$ is monic, it suffices to show $\mu_x p \phi A_s p = \mu_x p$, and for that we have

\begin{displaymath}
\mu_x p \phi A_s p = B_f q \phi A_s p = B_f \mu_y A_s p = B_f B_s \mu_x p = B_{f s} \mu_x p = \mu_x p.
\end{displaymath}
And for the second, we have

\begin{displaymath}
q \phi A_s p = \mu_y A_s p = B_s \mu_x p = B_s B_f q = B_{f s} q = q.
\end{displaymath}
\end{proof}
\begin{lemma}
\label{LatchSub}\hypertarget{LatchSub}{}
Let $R$ be elegant, $M$ a category with pullback-stable colimits, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$. Then for any object $x\in R$, the following square is a pullback, where $L_x$ denotes the Reedy latching object at $x$:

\begin{displaymath}
\itexarray{ L_x A & \to & A_x \\
  {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\
  L_x B & \to & B_x.  }
\end{displaymath}
\end{lemma}
\begin{proof}
The map $L_x B \to B_x$ is by definition the colimit in $M/B_x$ of a diagram whose objects are morphisms of the form $B_f : B_y \to B_x$, for $f$ a codegeneracy. By the Lemma \ref{DegSub}, each of these pulls back along $\mu_x$ to $A_f : A_y \to A_x$, forming the corresponding diagram whose colimit is $L_x A \to A_x$, and by assumption the pullback preserves the colimit.

\end{proof}
\hypertarget{all_presheaves_are_reedy_monomorphic}{}\subsubsection*{{All presheaves are ``Reedy monomorphic''}}\label{all_presheaves_are_reedy_monomorphic}

\begin{lemma}
\label{ReedyCofibrant}\hypertarget{ReedyCofibrant}{}
Let $R$ be elegant and let $M$ be an [[infinitary-coherent category]]. Then for any $x\in R$ and $A\in M^{R^{\mathrm{op}}}$, the map $L_x A \to A_x$ is a monomorphism.

\end{lemma}
\begin{proof}
We use the terminology from the page [[∞-ary exact category]]. Consider the [[sink]] with target $A_x$ consisting of all morphisms $A_f : A_y \to A_x$ indexed by nonidentity codegeneracies $f$ with domain $x$. By assumption, for any two such $f:x\to y$ and $f':x\to y'$ there is an absolute pushout $g:y\to z$ and $g':y'\to z$. By absoluteness, $A_z$ is the pullback $A_y \times_{A_x} A_y$. Thus, the images of these absolute pushouts form the \emph{kernel} of this sink.

Now $L_x A$ is the colimit of the diagram whose objects are $A_y$ indexed by such $f:x\to y$ and whose morphisms are $A_g: A_{y'} \to A_{y}$ for $g:y\to y'$ a codegeneracy with $g f = f'$. In this case, by the universal property of pullback, we have a unique map from $A_{y'}$ to $A_z$, where $z$ is the absolute pushout of $f$ and $f'$. Thus, a cocone under the above kernel is also a cocone under this diagram, and the converse is easy to see. Hence, $L_x A$ is the quotient of the above kernel.

However, in any infinitary-regular category, the quotient of the kernel of a sink is exactly the extremal-epic / monic factorization of that sink. Therefore, the induced map $L_x A \to A_x$ is monic.

\end{proof}
\hypertarget{reedy__injective}{}\subsubsection*{{Reedy = injective}}\label{reedy__injective}

\begin{theorem}
\label{RelativeLatching}\hypertarget{RelativeLatching}{}
If $R$ is elegant and $M$ is a [[Grothendieck topos]], then for any $x\in R$ and monomorphism $\mu:A\to B$ in $M^{R^{\mathrm{op}}}$, the induced map $L_x B \sqcup_{L_x A} A_x \to B_x$ is monic.

\end{theorem}
\begin{proof}
Since Grothendieck toposes are infinitary-coherent, by Lemma \ref{ReedyCofibrant} $L_x B\to B_x$ is monic. By assumption $A_x \to B_x$ is monic. And since Grothendieck toposes have pullback-stable colimits, by Lemma \ref{LatchSub} the square

\begin{displaymath}
\itexarray{ L_x A & \to & A_x \\
  {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\
  L_x B & \to & B_x.  }
\end{displaymath}
is a pullback. In other words, $L_x A$ is the [[intersection]] of the subobjects $L_x B$ and $A_x$ of $B_x$. But in any [[coherent category]], the pushout of two subobjects over their intersection is their [[union]], and hence in particular a subobject of their common codomain.

\end{proof}
\begin{ucor}
If $R$ is elegant and $M$ is a [[Cisinski model category]], then the [[Reedy model structure]] on $M^{R^{\mathrm{op}}}$ coincides with the [[injective model structure]].

\end{ucor}
\begin{proof}
By definition, they have the same [[weak equivalences]], so it suffices to show that their classes of cofibrations coincide. But every Reedy cofibration in any Reedy model structure is an injective (i.e. objectwise) cofibration, and the converse is Theorem \ref{RelativeLatching}.

\end{proof}
The most common application is when $M = SSet$. Thus, for instance, every simplicial presheaf on an elegant Reedy category is Reedy cofibrant.

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

\begin{itemize}%
\item The [[simplex category]] $\Delta$ is an elegant Reedy category.


\item Joyal's [[Theta category|disk categories]] $\Theta_n$ are elegant Reedy categories.


\item Every [[direct category]] is a Reedy category with no degeneracies, hence trivially an elegant one.


\item If $X$ is any presheaf on an elegant Reedy category $R$, then the opposite of its [[category of elements]] $(el X)^{op}$ is again an elegant Reedy category. This is fairly easy to see from the fact that $Set^{el X}$ is equivalent to the slice category $Set^{R^{op}}/X$.


\item Every [[EZ-Reedy category]] is elegant.



\end{itemize}
Note that unlike the notion of [[Reedy category]], the notion of elegant Reedy category is not self-dual: if $R$ is elegant then $R^{op}$ will not generally be elegant.

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

\begin{itemize}%
\item [[generalized Reedy category]]


\item [[Reedy model structure]]



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

\begin{itemize}%
\item [[Julie Bergner]] and [[Charles Rezk]], \emph{Reedy categories and the $\Theta$-construction}, \href{http://arxiv.org/abs/1110.1066}{arXiv:1110.1066}

\end{itemize}
Elegant Reedy categories are useful to model [[homotopy type theory]].

\begin{itemize}%
\item [[Mike Shulman]], \emph{The univalence axiom for elegant Reedy presheaves} (\href{http://arxiv.org/abs/1307.6248}{arXiv:1307.6248})


\item [[Michael Shulman]], \emph{Univalence for inverse diagrams and homotopy canonicity}, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( \emph{From type theory and homotopy theory to Univalent Foundations of Mathematics} ) June 2015 (\href{https://arxiv.org/abs/1203.3253}{arXiv:1203.3253}, \href{https://doi.org/10.1017/S0960129514000565}{doi:/10.1017/S0960129514000565})


\item Benno van den Berg and Ieke Moerdijk, \emph{W-types in homotopy type theory}, \href{http://www.staff.science.uu.nl/~berg0002/papers/WinHoTT.pdf}{PDF}



\end{itemize}
[[!redirects elegant Reedy categories]]



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