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\begin{document}

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\section*{Dugger&#39;s theorem}

\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{statement}{Statement}\dotfill \pageref*{statement} \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}

Dugger's theorem identifies [[combinatorial model categories]] as the [[model category]]-presentations of [[locally presentable (infinity,1)-categories]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Dugger's theorem)}

Every [[combinatorial model category]] $C$ is [[Quillen equivalence|Quillen equivalent]] to a left [[Bousfield localization of model categories|Bousfield localization]] $L_S SPSh(K)_{proj}$ of the global projective [[model structure on simplicial presheaves]] $SPSh(K)_{proj}$ on a [[small category]] $K$

\begin{displaymath}
L_S SPSh(K)_{proj} \stackrel{\simeq_{Quillen}}{\to}
  C
  \,.
\end{displaymath}
\end{theorem}
This is (\hyperlink{Dugger01}{Dugger 01, theorem 1.1}) building on results in (\hyperlink{DuggerUniversalHomotopy}{DuggerUniversalHomotopy}).

\begin{proof}
The proof proceeds (the way Dugger presents it, at least) in roughly three steps:

\begin{enumerate}%
\item Use that $[C^{op}, sSet_{Quillen}]_{proj}$ is in some precise sense the \emph{homotopy-} [[free cocompletion]] of $C$. This means that every functor $\gamma : C \to B$ from a plain category $C$ to a model category $B$ factors in an essentially unique way through the [[Yoneda embedding]] $j : C \to [C^{op},sSet]$ by a [[Quillen adjunction]]

\begin{displaymath}
(\hat \gamma \dashv R) : 
  B
  \stackrel{\overset{\hat \gamma}{\leftarrow}}
   {\underset{R}{\to}}
  [C^{op}, sSet_{Quillen}]_{proj}  
  \,.
\end{displaymath}
The detailed definitions and detailed proof of this are discussed at [[(∞,1)-category of (∞,1)-presheaves]].


\item For a given combinatorial model category $B$, choose $C := B_\lambda^{cof}$ the full [[subcategory]] on a [[small set]] (guaranteed to exist since $B$ is [[locally presentable category|locally presentable]]) of cofibrant $\lambda$-[[compact object]]s, for some [[regular cardinal]] $\lambda$, and show that the induced Quillen adjunction

\begin{displaymath}
B \stackrel{\overset{\hat i}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  [(B_\lambda^{cof})^{op}, sSet]_{proj}
\end{displaymath}
induced by the above statement from the inclusion $i : B_\lambda^{cof} \hookrightarrow B$ exhibits $B$ as a homotopy-[[reflective subcategory]] in that the [[derived adjunction counit]] $\hat i \circ Q \circ R \stackrel{\simeq}{\to} Id$ ($Q$ some cofibrant replacement functor) is a [[natural transformation|natural]] [[weak equivalence]] on fibrant objects (recall from [[adjoint functor]] the characterization of adjoints to full and faithful functors).


\item Define $S$ to be the set of morphisms in $[(C_\lambda^{cof})^{op}, sSet]$ that the left [[derived functor]] $\hat i \circ Q$ of $\hat i$ (here $Q$ is some cofibrant replacement functor) sends to weak equivalences in $B$. Then form the left [[Bousfield localization of model categories|Bousfield localization]] $L_S [(C_\lambda^{cof})^{op},sSet]_{proj}$ with respect to this set of morphisms and prove that this is [[Quillen equivalence|Quillen equivalent]] to $B$.



\end{enumerate}
Carrying this program through requires the following intermediate results.

First recall from the discussion at [[(∞,1)-category of (∞,1)-presheaves]] that to produce the Quilen adjunction $(\hat i \dashv R)$ from $i$, we are to choose a [[cofibrant resolution]] functor

\begin{displaymath}
I : C \to [\Delta,B]
\end{displaymath}
of $i : C= B_\lambda^{cof} \to B$.

The [[adjunct]] of this is a functor $\tilde I : C \times \Delta \to B$. For each object $b \in B$ write $(C \times \Delta \downarrow b)$ for the [[slice category]] induced by this functor.

\textbf{Lemma} (Dugger, prop. 4.2)

For every fibrant object $b \in B$ we have that the [[homotopy colimit]] $hocolim (C \times \Delta \downarrow b) \to B)$ is weakly equivalent to $\hat i \circ Q\circ R (b)$.

\textbf{Corollary} (Dugger, cor. 4.4) The induced Quillen adjunction

\begin{displaymath}
B \stackrel{\leftarrow}{\to}
  [C^{op}, sSet]
\end{displaymath}
is a homotopy-reflective embedding precisely if the canonical morphisms

\begin{displaymath}
hocolim (C \times \Delta \downarrow b) \to b
\end{displaymath}
are weak equivalences for every fibrant object $b \in B$.

\ldots{}

\end{proof}
Notice that the theorem just mentions plain combinatorial model categories, not [[simplicial model category|simplicial model categories]]. But of course by basic facts of [[enriched category theory]] $Funct(C^{op}, SSet)$ is an [[SSet]]-[[enriched category]] and its projective [[global model structure on functors]] $Func(C^{op}, SSet)_{proj}$ is compatibly a [[simplicial model category]], as are all its [[Bousfield localization of model categories|Bousfield localizations]]. (See [[model structure on simplicial presheaves]] for more details.) Therefore an immediate but very useful corollary of the above statement is

\begin{corollary}
\label{}\hypertarget{}{}
Every combinatorial model category is [[Quillen equivalence|Quillen equivalent]] to one which is

\begin{itemize}%
\item a [[simplicial model category]]


\item a left [[proper model category]].



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

[[!include locally presentable categories - table]]

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

Dugger's theorem is due to

\begin{itemize}%
\item [[Daniel Dugger]], \emph{[[Combinatorial model categories have presentations]]}, Adv. Math. 164 (2001), no. 1, 177-201 (\href{http://arxiv.org/abs/math/0007068}{arXiv:math/0007068})

\end{itemize}
based on results in

\begin{itemize}%
\item [[Dan Dugger]], \emph{[[Universal homotopy theories]]}

\end{itemize}
The interpretation in terms of [[locally presentable (infinity,1)-categories]] is due to

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}


\end{document}