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\begin{document}

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\section*{model structure on dg-categories}

\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{with_dwyerkan_weak_equivalences}{With Dwyer-Kan weak equivalences}\dotfill \pageref*{with_dwyerkan_weak_equivalences} \linebreak
\noindent\hyperlink{WithMoritaEquivalences}{With Morita equivalences}\dotfill \pageref*{WithMoritaEquivalences} \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 [[model structure on enriched categories]] gives in particular a model structure on dg-categories, called the \emph{Dwyer-Kan model structure}, which is analogous to the usual [[model structure on sSet-categories]] which models [[(infinity,1)-categories]].

There are interesting [[left Bousfield localizations]] of this model structure, called the \emph{quasi-equiconic} and \emph{Morita} model structures. Here the [[fibrant objects]] are the [[pretriangulated dg-categories]], resp. [[idempotent complete category|idempotent complete]] [[pretriangulated dg-categories]]. In characteristic zero, the Morita model structure is known to present the [[(infinity,1)-category]] of linear [[stable (infinity,1)-categories]] (\hyperlink{Cohn13}{Cohn 13}).

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

\hypertarget{with_dwyerkan_weak_equivalences}{}\subsubsection*{{With Dwyer-Kan weak equivalences}}\label{with_dwyerkan_weak_equivalences}

\begin{theorem}
\label{}\hypertarget{}{}
Let $k$ be a [[commutative ring]]. Write $dgCat_k$ for the [[category]] of [[small category|small]] [[dg-categories]] over $k$.

There is the structure of a [[cofibrantly generated model category]] on $dgCat_k$ where a dg-functor $F : A \to B$ is

\begin{itemize}%
\item a weak equivalence if

\begin{enumerate}%
\item for all objects $x,y \in A$ the component $F_{x,y} : A(x,y) \to B(F(x), F(y))$ is a [[quasi-isomorphism]] of [[chain complexes]];


\item the induced functor on [[homotopy categories]] $H^0(F)$ (obtained by taking degree 0 [[chain homology]] in each hom-object) is an [[equivalence of categories]].



\end{enumerate}

\item a fibration if

\begin{enumerate}%
\item for all objects $x,y \in A$ the component $F_{x,y}$ is a degreewise surjection of chain complexes;


\item for each [[isomorphism]] $F(x) \to Z$ in $H^0(B)$ there is a lift to an isomorphism in $H^0(A)$.



\end{enumerate}


\end{itemize}
\end{theorem}
This is due to (\hyperlink{Tabuada}{Tabuada}).

\begin{remark}
\label{}\hypertarget{}{}
The definition is entirely analogous to the [[model structure on sSet-categories]]. Both are special cases of the [[model structure on enriched categories]].

\end{remark}
\hypertarget{WithMoritaEquivalences}{}\subsubsection*{{With Morita equivalences}}\label{WithMoritaEquivalences}

There is another model category structure with more weak equivalences, the \emph{Morita equivalences} (\hyperlink{Tabuada05}{Tabuada 05}). This is in fact the [[left Bousfield localization]] of the above model structure with respect to the Morita equivalences, i.e. functors $F: C \to D$ whose induced [[restriction of scalars]] functor $\mathbf Lf^* : \mathbf D(D) \to \mathbf D(C)$ is an [[equivalence of categories]].

The [[fibrant objects]] with respect to this model structure are the [[dg-categories]] A for which the canonical inclusion $H^0(A) \hookrightarrow \mathbf D(A)$ has its [[essential image]] stable under [[cones]], [[suspensions]], and [[direct sums]]. Hence the [[homotopy category]] with respect to this model structure is identified with the [[full subcategory]] of Ho(DGCat), the homotopy category of the Dwyer-Kan model structure, spanned by dg-categories of this form.

This model structure is a presentation of the [[(∞,1)-category]] of [[stable (∞,1)-categories]] (\hyperlink{Cohn13}{Cohn 13}).

The [[pretriangulated dg-category|pretriangulated envelope]] of [[Bondal]]-[[Kapranov]] is a [[fibrant replacement]] functor for the Morita model structure. The [[DG quotient]] of [[Drinfeld]] is a model for the [[homotopy cofibre]] with respect to the Morita model structure.

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

\begin{itemize}%
\item [[model structure on dg-operads]]


\item [[model structure on dg-algebras over an operad]]

\begin{itemize}%
\item [[model structure on dg-algebras]]


\item \textbf{model structure on dg-categories}



\end{itemize}


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

The model structure on dg-categories is due to

\begin{itemize}%
\item [[Gonçalo Tabuada]], \emph{Une structure de cat\'e{}gorie de mod\`e{}les de Quillen sur la cat\'e{}gorie des dg-cat\'e{}gories} C. R. Acad. Sci. Paris S\'e{}r. I Math. 340 (1) (2005), 15--19.

\end{itemize}
It is reproduced as theorem 4.1 in

\begin{itemize}%
\item [[Bernhard Keller]], \emph{On differential graded categories} (\href{http://atlas.mat.ub.es/grgta/articles/Keller.pdf}{pdf})

\end{itemize}
Discussion of [[internal homs]] of dg-categories in terms of refined [[Fourier-Mukai transforms]] is in

\begin{itemize}%
\item [[Bertrand Toën]], \emph{The homotopy theory of dg-categories and derived Morita theory}, Invent. Math. 167 (2007), 615--667

\end{itemize}
Discussion of [[internal homs]] of dg-categories using (just) the structure of a [[category of fibrant objects]] is in

\begin{itemize}%
\item Alberto Canonaco, Paolo Stellari, \emph{Internal Homs via extensions of dg functors} (\href{http://arxiv.org/abs/1312.5619}{arXiv:1312.5619})

\end{itemize}
The derived internal Hom in the homotopy category of DG-categories is equivalent to the dg-category of A\_infty-functors.

\begin{itemize}%
\item [[Giovanni Faonte]], \emph{A-infinity functors and homotopy theory of DG-categories}, \href{http://arxiv.org/abs/1412.1255}{arXiv}.

\end{itemize}
A proof that the [[internal hom]] of Ho(DGCat) constructed by To\"e{}n is in fact the right [[derived functor]] of the internal hom of DGCat is in

\begin{itemize}%
\item [[Beatriz Rodriguez Gonzalez]], \emph{A derivability criterion based on the existence of adjunctions}, 2012, \href{http://arxiv.org/abs/1202.3359}{arXiv:1202.3359}.

\end{itemize}
There is also

\begin{itemize}%
\item David Rosoff, \emph{Mapping spaces of $A_\infty$-algebras} (\href{http://www.math.washington.edu/~rosoff/ainfs.pdf}{pdf})

\end{itemize}
The model structure with Morita equivalences as weak equivalences is discussed in

\begin{itemize}%
\item [[Goncalo Tabuada]], \emph{Invariants additifs de dg-catgories}. Internat. Math. Res. Notices 53 (2005), 33093339.

\end{itemize}
That the Morita model structure on dg-categories presents the homotopy theory of $k$-linear [[stable (infinity,1)-categories]] was shown in

\begin{itemize}%
\item [[Lee Cohn]], \emph{Differential Graded Categories are k-linear Stable Infinity Categories} (\href{http://arxiv.org/abs/1308.2587}{arXiv:1308.2587})

\end{itemize}


\end{document}