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\begin{document}

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\section*{dg-category}

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

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_infinity1category_of_dgcategories}{The (infinity,1)-category of dg-categories}\dotfill \pageref*{the_infinity1category_of_dgcategories} \linebreak
\noindent\hyperlink{relation_to_stable_categories}{Relation to stable $\infty$-categories}\dotfill \pageref*{relation_to_stable_categories} \linebreak
\noindent\hyperlink{aspects_of_dgcategories}{Aspects of dg-categories}\dotfill \pageref*{aspects_of_dgcategories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{overviews}{Overviews}\dotfill \pageref*{overviews} \linebreak
\noindent\hyperlink{homotopy_theory_of_dgcategories}{Homotopy theory of dg-categories}\dotfill \pageref*{homotopy_theory_of_dgcategories} \linebreak
\noindent\hyperlink{derived_noncommutative_geometry}{Derived noncommutative geometry}\dotfill \pageref*{derived_noncommutative_geometry} \linebreak
\noindent\hyperlink{other_aspects}{Other aspects}\dotfill \pageref*{other_aspects} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Differential graded categories} or \emph{dg-categories} are linear analogues of [[spectral categories]]. In other words they are [[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]]. It is common and useful to view them as [[enhanced triangulated categories]].

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

A \emph{dg-category} over a [[commutative ring]] $k$ is an [[(infinity,1)-category]] [[enriched (infinity,1)-category|enriched]] in the [[(infinity,1)-category of chain complexes]] of $k$-[[modules]]. Equivalently, it is an ordinary category [[enriched category|strictly enriched]] in [[chain complexes]] (see \hyperlink{Haugseng13}{Haugseng 13}).

Hence a dg-category is a category with [[mapping complexes]] of morphisms between any two objects. By taking the [[homologies]] of these [[chain complexes]] in degree zero, one gets an ordinary category, called the [[homotopy category of a dg-category]]. Notice that a dg-category with a single object is the same thing as a [[dg-algebra]].

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

\hypertarget{the_infinity1category_of_dgcategories}{}\subsubsection*{{The (infinity,1)-category of dg-categories}}\label{the_infinity1category_of_dgcategories}

The [[Dwyer-Kan model structure on dg-categories]] presents the [[(infinity,1)-category of dg-categories]].

\hypertarget{relation_to_stable_categories}{}\subsubsection*{{Relation to stable $\infty$-categories}}\label{relation_to_stable_categories}

By the [[stable Dold-Kan correspondence]], the [[(infinity,1)-category]] of dg-categories is equivalent to the [[(infinity,1)-category]] of [[(infinity,1)-categories]] [[enriched (infinity,1)-category|enriched]] in the [[symmetric monoidal (infinity,1)-category]] of [[module spectra|modules]] over the [[Eilenberg-Mac Lane spectrum]] $H k$. The latter is equivalent, at least morally, to the [[(infinity,1)-category]] of $k$-[[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]].

More precisely, it is shown in \hyperlink{Cohn13}{Cohn 13} that the [[Morita model structure on dg-categories]] presents the [[(infinity,1)-category]] of [[idempotent complete (infinity,1)-category|idempotent complete]] [[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]].

\hypertarget{aspects_of_dgcategories}{}\subsection*{{Aspects of dg-categories}}\label{aspects_of_dgcategories}

\begin{itemize}%
\item [[homotopy category of a dg-category]].
\item [[equivalence of dg-categories]]
\item [[dg-modules]], [[perfect dg-modules]]
\item [[derived dg-category]]
\item [[dg-Yoneda embedding]]
\item [[pretriangulated dg-category]]
\item [[dg-localization]], [[dg-quotient]]
\item [[dg-nerve]]
\item [[Waldhausen K-theory of a dg-category]]
\item [[semi-topological K-theory of a dg-category]]
\item [[derived moduli stack of objects in a dg-category]]

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

\begin{itemize}%
\item [[enhanced triangulated category]]
\item [[A-infinity category]]
\item [[spectral category]]
\item [[stable (infinity,1)-category]]

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

Historically, dg-categories were introduced in

\begin{itemize}%
\item [[G. M. Kelly]], \emph{Chain maps inducing zero homology maps}, Proc. Cambridge Philos. Soc. 61 (1965), 847--854,

\end{itemize}
whilst their modern development can be traced to

\begin{itemize}%
\item [[A. I. Bondal]], [[Mikhail Kapranov]], \emph{Enhanced triangulated categories}, . ,  181 (1990), No.5, 669--683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93--107, (MR91g:18010) ([[bondalKaprEnhTRiangCat.pdf:file]])

\end{itemize}
\hypertarget{overviews}{}\subsubsection*{{Overviews}}\label{overviews}

For concise reviews of the theory, see section 1 of

\begin{itemize}%
\item [[A. Beilinson]], [[V. Vologodsky]], \emph{DG guide to Voevodsky's motives}.

\end{itemize}
as well as the introduction and appendices to

\begin{itemize}%
\item [[V. Drinfeld]], \emph{DG quotients of DG categories}, \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/DGquotients.pdf}{pdf}.

\end{itemize}
For longer surveys, see

\begin{itemize}%
\item [[Bernhard Keller]], \emph{On differential graded categories} International Congress of Mathematicians. Vol. II, 151--190, Eur. Math. Soc., Z\"u{}rich, 2006. (\href{http://arxiv.org/abs/math/0601185}{arXiv})

\end{itemize}
and

\begin{itemize}%
\item [[Bertrand Toën]], \emph{Lectures on dg-categories} (\href{https://perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf}{pdf}).

\end{itemize}
\hypertarget{homotopy_theory_of_dgcategories}{}\subsubsection*{{Homotopy theory of dg-categories}}\label{homotopy_theory_of_dgcategories}

The [[homotopy theory]] of [[dg-categories]] is studied in

\begin{itemize}%
\item [[Gonçalo Tabuada]], \emph{Homotopy theory of DG categories}, Thesis, Paris, 2007, \href{http://people.math.jussieu.fr/~keller/TabuadaThese.pdf}{pdf}.


\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. Math. Acad. Sci. Paris 340 (2005), no. 1, 15--19.



\end{itemize}
The equivalence with the [[homotopy theory]] of [[stable (infinity,1)-categories]] is discussed 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}
(Note that the proof works over any ring, even though it is stated there for [[characteristic zero]].)

In the following it is shown that the [[homotopy theory]] of [[(infinity,1)-categories]] enriched in the [[(infinity,1)-category]] of [[chain complexes]] is equivalent to the [[homotopy theory]] of ordinary categories strictly enriched in [[chain complexes]].

\begin{itemize}%
\item [[Rune Haugseng]], \emph{Rectification of enriched infinity-categories}, \href{http://arxiv.org/abs/1312.3881v2}{arXiv:1312.3881}.

\end{itemize}
\hypertarget{derived_noncommutative_geometry}{}\subsubsection*{{Derived noncommutative geometry}}\label{derived_noncommutative_geometry}

The following references discuss the use of dg-categories in [[derived noncommutative algebraic geometry]] and [[noncommutative motives]].

\begin{itemize}%
\item [[Gonçalo Tabuada]], \emph{Invariants additifs de DG-cat\'e{}gories}, Int. Math. Res. Not. 2005, no. 53, 3309--3339; Addendum in Int. Math. Res. Not. 2006, Art. ID 75853, 3 pp. ; Erratum in Int. Math. Res. Not. IMRN 2007, no. 24, Art. ID rnm149, 17 pp.


\item [[Marco Robalo]], \emph{Th\'e{}orie homotopique motivique des espaces noncommutatifs}, \href{http://webusers.imj-prg.fr/~marco.robalo/these.pdf}{pdf}.


\item S. Mahanta, \emph{Noncommutative geometry in the framework of differential graded categories}, ()


\item D. Orlov, \emph{Smooth and proper noncommutative schemes and gluing of DG categories}, Adv. Math. 302 (2014) \href{https://dx.doi.org/10.1016/j.aim.2016.07.014}{doi}



\end{itemize}
\hypertarget{other_aspects}{}\subsubsection*{{Other aspects}}\label{other_aspects}

\begin{itemize}%
\item [[Bernhard Keller]], \emph{Deriving DG categories}, Ann. Sci. \'E{}cole Norm. Sup. (4) 27 (1994), no. 1, 63--102 ()


\item [[Dmitry Tamarkin]], \emph{What do dg-categories form?}, Compos. Math. 143 (2007), no. 5, 1335--1358.


\item [[Michael Batanin|M. Batanin]], \emph{What do dg-categories form} (after Tamarkin), talks at Paris 7 and Australian category seminar (), \href{http://arxiv.org/abs/math.CT/0606553}{math.CT/0606553}


\item [[Oren Ben-Bassat]], [[Jonathan Block]], \emph{Cohesive DG categories I: Milnor descent}, \href{http://arxiv.org/abs/1201.6118}{arxiv/1201.6118}



\end{itemize}
[[!redirects dg-category]] [[!redirects dg-categories]] [[!redirects dg category]] [[!redirects dg categories]] [[!redirects DG-category]] [[!redirects DG-categories]] [[!redirects DG category]] [[!redirects DG categories]] [[!redirects differential graded category]] [[!redirects differential graded categories]]



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