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\begin{document}

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\section*{projectively cofibrant diagram}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_specific_diagram_shapes}{For specific diagram shapes}\dotfill \pageref*{for_specific_diagram_shapes} \linebreak
\noindent\hyperlink{for_specific_ambient_model_categories}{For specific ambient model categories}\dotfill \pageref*{for_specific_ambient_model_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Let $\mathcal{C}$ be a ([[cofibrantly generated model category|cofibrantly generated]]) [[model category]] and let $\mathcal{D}$ be any [[category]], regarded as a [[diagram]]-shape in the following.

Write $[\mathcal{D}, \mathcal{C}]_{proj}$ for the [[projective model structure]] on the [[functor category]] of [[functors]] from $\mathcal{D}$ to $\mathcal{C}$, hence of $\mathcal{D}$-[[diagrams]] in $\mathcal{C}$.

\begin{defn}
\label{ProjectivelyCofibrantDiagram}\hypertarget{ProjectivelyCofibrantDiagram}{}
A functor/diagram $X : \mathcal{D} \to \mathcal{C}$ is a \textbf{projectively cofibrant diagram} in $\mathcal{C}$ if it is a [[cofibrant object]] in the [[projective model structure]] $[\mathcal{D}, \mathcal{C}]_{proj}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
We unwind the condition in def. \ref{ProjectivelyCofibrantDiagram}.

First of all it says of course that a [[diagram]] $X \colon \mathcal{D}\to \mathcal{C}$ is projectively cofibrant precisely if the inclusion $\emptyset \to X$ of the [[initial object|initial]] diagram has the [[left lifting property]] with respect to [[natural transformations]] of diagrams

\begin{displaymath}
(A \stackrel{p}{\to} B) : \mathcal{C} \to \mathcal{D}
\end{displaymath}
which are projective acyclic fibrations, hence which are such that for each $c \in \mathcal{C}$ the component $\eta_c : A(c) \to B(c)$ is an acyclic fibration in $\mathcal{C}$.

This in turn means that $F$ is projectively cofibrant precisely if for every diagram of natural transformations

\begin{displaymath}
\itexarray{
  &&A
  \\
  &
  &\downarrow^{\mathrlap{p}}
  \\
  X
  &\to&
  B
}
\end{displaymath}
with $p$ as above, there exists a lift $\sigma$ in

\begin{displaymath}
\itexarray{
  &&A
  \\
  &
  {}^{\mathllap{\sigma}}\nearrow &\downarrow^{\mathrlap{p}}
  \\
  X
  &\to&
  B
}
 \;\;\;\;\;\;
 \in [\mathcal{D}, \mathcal{C}]
  \,.
\end{displaymath}
making the triangle commute.

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

The main point of projectively cofibrant diagrams is that the ordinary [[colimit]] over them is a presentation of the [[homotopy colimit]]:

because the ([[colimit]] $\dashv$ constant diagram)-[[adjunction]]

\begin{displaymath}
(\underset{\longrightarrow}{\lim} \dashv const)
  :
  \mathcal{C} \stackrel{\overset{\underset{\longrightarrow}{\lim}}{\leftarrow}}{\underset{const}{\to}}
  [\mathcal{C}, \mathcal{D}]
\end{displaymath}
is a [[Quillen adjunction]] (because $const$ is by the very definition of the [[projective model structure]] a [[right Quillen functor]]), the [[homotopy colimit]], being the left [[derived functor]] $\mathbb{L}\underset{\longrightarrow}{\lim}$ of the [[colimit]], is computed as the ordinary colimit evaluated on a cofibrant resolution $Q X$ of a diagram $X : \mathcal{D} \to \mathcal{C}$:

\begin{displaymath}
(\mathbb{L} \underset{\longrightarrow}{\lim})(X)
  \simeq
  \underset{\longrightarrow}{\lim})(Q X)
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{for_specific_diagram_shapes}{}\subsubsection*{{For specific diagram shapes}}\label{for_specific_diagram_shapes}

\begin{example}
\label{}\hypertarget{}{}
A \emph{[[span]]} diagram $X_1 \leftarrow X_0 \to X_1$ is projectively cofibrant precisely if the two morphisms are cofibrations in $\mathcal{D}$ and $X_0$, hence all three objects, are cofibrant.

The colimit over such a diagram is the [[homotopy pushout]] of the span.

\end{example}
\begin{example}
\label{CofibrantCotowerDiagram}\hypertarget{CofibrantCotowerDiagram}{}
A \emph{cotower} diagram

\begin{displaymath}
X_0 \to X_1 \to X_2 \to \cdots
\end{displaymath}
is projectively cofibrant precisely if every morphism is a cofibration and if the first object $X_0$, and hence all objects, are cofibrant in $\mathcal{D}$.

The colimit over such a diagram is a homotopy [[sequential colimit]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A \emph{[[parallel morphisms]]} diagram

\begin{displaymath}
X_0 \stackrel{\overset{f}{\to}}{\underset{g}{\to}} X_1
\end{displaymath}
is projectively cofibrant precisely if $X_0$ is cofibrant, and if the morphism $(f,g) : X_0 \coprod X_0 \to X_1$ is a cofibration.

This implies that also $f$ and $g$ are cofibrations and hence that $X_1$ is cofibrant.

\end{example}
The colimit over such a diagram is a homotopy [[coequalizer]].

\hypertarget{for_specific_ambient_model_categories}{}\subsubsection*{{For specific ambient model categories}}\label{for_specific_ambient_model_categories}

Let $\mathcal{C} =$ [[sSet]]${}_{Quillen}$ be the standard [[model structure on simplicial sets]]. Then $[\mathcal{D}, \mathcal{C}]_{proj}$ is the projective [[model structure on simplicial presheaves]].

For the following see at \emph{[[model structure on simplicial presheaves]]} the section \emph{\href{model+structure%20on%20simplicial%20presheaves#CofibrantObjects}{Cofibrant objects}} for more details (due to [[Dan Dugger]]).

\begin{prop}
\label{}\hypertarget{}{}
A sufficient condition for a diagram $X : \mathcal{D} \to sSet$ to be projectively cofibrant is:

\begin{enumerate}%
\item $X$ is degreewise a coproducts of [[representable functor|representables]]

\begin{displaymath}
X_n = \coprod_{i} U^n_i \;\;\;\;  \{U^n_i \in \mathcal{C} \hookrightarrow [\mathcal{C}, Set]\}
\end{displaymath}

\item the degenerate cells in each degree form a separate [[coproduct]] summand;

\begin{displaymath}
X_n = NonDegenerate \coprod Degenerate
   \,.
\end{displaymath}


\end{enumerate}
\end{prop}
\begin{example}
\label{}\hypertarget{}{}
A \emph{[[split hypercover]]} is of this form.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
For $X : \mathcal{D} \to sSet$ any simplicial presheaf, a cofibrant [[resolution]] is given by

\begin{displaymath}
(Q X)_n : \coprod_{U_0 \to \cdots \to U_n \to X_n} U_0
  \,,
\end{displaymath}
where the coproduct runs over all sequences of morphisms between representables $U_i$, as indicated.

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

See the references at \emph{[[homotopy colimit]]} and generally at \emph{[[model category]]}.

Related discussion is at

\begin{itemize}%
\item MathOverflow \emph{\href{http://mathoverflow.net/questions/70612/when-are-diagrams-of-cofibrations-projectively-cofibrant}{When are ``diagrams of cofibrations'' projectively cofibrant}}

\end{itemize}
Related discussion is for instance also in

\begin{itemize}%
\item Urs Schreiber, \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
where cofibrant cotowers are mentioned as example 2.3.15.

[[!redirects projectively cofibrant diagrams]]



\end{document}