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\begin{document}

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\section*{pretriangulated dg-category}

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

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

[[!include homological algebra - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{pretriangulated_dgcategories}{Pretriangulated dg-categories}\dotfill \pageref*{pretriangulated_dgcategories} \linebreak
\noindent\hyperlink{strongly_pretriangulated_dgcategories}{Strongly pretriangulated dg-categories}\dotfill \pageref*{strongly_pretriangulated_dgcategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \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}

\emph{Pretriangulated dg-categories} over a [[commutative ring]] $k$ are, roughly speaking, [[dg-categories]] whose [[homotopy category of a dg-category|homotopy category]] is canonically [[triangulated category|triangulated]]. These form a model for [[stable (∞,1)-category|stable]] [[k-linear (∞,1)-categories]], in a sense which is made precise below (at least in [[characteristic zero]]). In other words pretriangulated dg-categories can be viewed as [[enhanced triangulated categories]]. For this reason some authors call them \emph{stable dg-categories}.

\hypertarget{history}{}\subsection*{{History}}\label{history}

The notion of pretriangulated dg-category goes back to \hyperlink{BondalKapranov90}{(Bondal-Kapranov 1990)}.

[[Goncalo Tabuada]] demonstrated the existence of a [[model structure]] on the category of small [[dg-categories]], the [[quasi-equiconic model structure on dg-categories]], where the [[fibrant objects]] are the pretriangulated dg-categories. See \hyperlink{Tabuada07}{(Tabuada 07, Theorem 2.2 and Proposition 2.10)}. This model structure can be [[left Bousfield localization|Bousfield localized]] to the [[Morita model structure on dg-categories]], where the [[fibrant objects]] are the [[idempotent complete (infinity,1)-category|idempotent complete]] pretriangulated dg-categories. In \hyperlink{Cohn13}{(Cohn 13)} it is shown that the associated [[(infinity,1)-category]] is equivalent to the [[(infinity,1)-category]] of [[stable (infinity,1)-category|stable]] [[k-linear (∞,1)-categories]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{pretriangulated_dgcategories}{}\subsubsection*{{Pretriangulated dg-categories}}\label{pretriangulated_dgcategories}

Let $A$ be a [[dg-category]] and $P(A)$ the dg-category of [[dg-presheaves]] or right [[dg-modules]] over $A$. The [[Yoneda embedding]] induces a [[fully faithful functor]] $h : ho(A) \hookrightarrow ho(P(A))$ on the [[homotopy category of a dg-category|homotopy categories]]. The category $ho(P(A))$ has a canonical [[triangulated structure]] (which can be written down directly).

\begin{udefn}
The dg-category $A$ is called \textbf{pretriangulated} if the functor $h$ is stable under the [[suspension functor]] (and its [[loop space functor|inverse]]), and under taking [[mapping cones]] in $ho(P(A))$.

\end{udefn}
\begin{uprop}
A dg-category $A$ is pretriangulated if and only if it is a [[fibrant object]] in the [[quasi-equiconic model structure on dg-categories]].

\end{uprop}
See \hyperlink{Tabuada07}{(Tabuada 07, Proposition 2.10)}.

\hypertarget{strongly_pretriangulated_dgcategories}{}\subsubsection*{{Strongly pretriangulated dg-categories}}\label{strongly_pretriangulated_dgcategories}

Let $A$ be a [[dg-category]].

\begin{udefn}
The \textbf{$n$-translation} of an object $X \in A$ is an object $X[n] \in A$ representing the functor

\begin{displaymath}
\Hom(\cdot, X)[n].
\end{displaymath}
The \textbf{cone} of a closed morphism $f : X \to Y$ of degree zero is an object $\Cone(f) \in A$ representing the functor

\begin{displaymath}
\Cone(\Hom(\cdot, X) \stackrel{f_*}{\to} \Hom(\cdot, Y)),
\end{displaymath}
which is a [[mapping cone]] in [[category of chain complexes|chain complexes]].

\end{udefn}
\begin{udefn}
The dg-category $A$ is called \textbf{strongly pretriangulated} if it admits a [[zero object]], all translations of all objects, and all cones of all morphisms.

\end{udefn}
\begin{udefn}
Let $A$ be a dg-category. A \textbf{strongly pretriangulated envelope} of $A$ is the data of a strongly pretriangulated dg-category $tri(A)$ and a [[fully faithful]] functor $A \hookrightarrow tri(A)$ such that any functor $u: A \to B$ to a strongly pretriangulated dg-category $B$ factors uniquely through a functor $tri(A) \to B$.

\end{udefn}
A strongly pretriangulated envelope $A \hookrightarrow tri(A)$ always exists, and may be constructed by taking $tri(A)$ to be the full dg-subcategory of $P(A)$ spanned by the objects of the [[full subcategory|full]] [[triangulated subcategory]] of $ho(P(A))$ generated by the [[representable presheaves]], and $A \hookrightarrow tri(A)$ to be the functor induced by the Yoneda embedding. There is also another construction using twisted complexes, see \hyperlink{BondalKapranov90}{Bondal-Kapranov}.

Now we have the following characterization of pretriangulated dg-categories.

\begin{prop}
\label{}\hypertarget{}{}
Let $A$ be a dg-category and $A \hookrightarrow tri(A)$ be a strongly pretriangulated envelope of $A$. $A$ is pretriangulated if and only if the induced fully faithful functor $ho(A) \hookrightarrow ho(tri(A))$ is [[essentially surjective]] (and hence an [[equivalence of categories]]).

\end{prop}
As an immediate corollary, note that for a pretriangulated dg-category $A$, its [[homotopy category of a dg-category|homotopy category]] $ho(A)$ inherits a canonical triangulated structure.

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

\begin{prop}
\label{}\hypertarget{}{}
Let $u : A \to B$ be a functor between two dg-categories. If $A$ and $B$ are pretriangulated then the induced functor $ho(u): ho(A) \to ho(B)$ is [[triangulated functor|triangulated]]. Further, $u$ is a [[quasi-equivalence of dg-categories|quasi-equivalence]] if and only if $ho(u)$ is a triangulated equivalence.

\end{prop}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

The use of pretriangulated dg-categories as [[enhanced triangulated categories]] has been especially successful in the study of the various [[triangulated categories of sheaves]] on [[algebraic varieties]].

In the below paper it is shown that the triangulated categories of [[quasicoherent sheaf|quasicoherent sheaves]] on quasiprojective varieties and some of their cousins (like categories of perfect complexes on quasiprojective varieties) have unique dg-enhancements. [[Fernando Muro]] has developed an obstruction theory for the existance and measuring non-uniqueness of dg-enhancements in more general settings (unpublished).

\begin{itemize}%
\item [[Valery Lunts|V. A. Lunts]], [[Dmitri Orlov|D. O. Orlov]], \emph{Uniqueness of enhancement for triangulated categories}, J. Amer. Math. Soc. \textbf{23} (2010), 853-908, \href{http://www.ams.org/journals/jams/2010-23-03/S0894-0347-10-00664-8/home.html}{journal}, \href{http://arxiv.org/abs/0908.4187}{arXiv:0908.4187}.


\item [[Valery Lunts]], Olaf M. Schnuerer, \emph{New enhancements of derived categories of coherent sheaves and applications}, 2014, \href{http://arxiv.org/abs/1406.7559}{arXiv}.



\end{itemize}
Similarly, pretriangulated dg-categories have proven to give a good model for [[derived noncommutative algebraic geometry]] in the sense of [[Maxim Kontsevich]]. See there for relevant references. In this connection see also the work of [[Goncalo Tabuada]] who has developed a theory of [[noncommutative motives]] in this framework.

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

\begin{itemize}%
\item [[triangulated category]]


\item [[enhanced triangulated category]]


\item [[stable derivator]]


\item [[stable (∞,1)-category]]



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

\begin{itemize}%
\item [[Alexei 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}
The [[model structure]] presenting pretriangulated dg-categories is discussed in

\begin{itemize}%
\item [[Goncalo Tabuada]], \emph{Theorie homotopique des DG-categories}, Ph.D. thesis, Universite Denis Diderot - Paris 7, \href{http://arxiv.org/abs/0710.4303}{arXiv:0710.4303}.

\end{itemize}
See also paragraph 2.3 of

\begin{itemize}%
\item [[Dmitri Orlov]], \emph{Smooth and proper noncommutative schemes and gluing of DG categories}, \href{http://arxiv.org/abs/1402.7364}{arXiv:1402.7364}.

\end{itemize}
For a summary of the various [[model structures on dg-categories]], see Section 2 of the paper

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Goncalo Tabuada]], \emph{Non-connective K-theory via universal invariants}, Compositio Math. 147 (2011), 1281-1320, \href{http://arxiv.org/abs/0903.3717}{arXiv:0903.3717}, \href{http://www.math.univ-toulouse.fr/~dcisinsk/Non-connective-K-theory.pdf}{pdf}.

\end{itemize}
The relation to [[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}
[[!redirects pretriangulated dg-categories]] [[!redirects strongly pretriangulated dg-category]] [[!redirects strongly pretriangulated dg-categories]] [[!redirects pre-triangulated dg-category]] [[!redirects pre-triangulated dg-categories]] [[!redirects stable dg-category]] [[!redirects stable dg-categories]]



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