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\begin{document}

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\section*{Deligne tensor product of abelian categories}

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

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_terms_of_categories_of_modules_and_tensor_product_of_algebras}{In terms of categories of modules and tensor product of algebras}\dotfill \pageref*{in_terms_of_categories_of_modules_and_tensor_product_of_algebras} \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 natural [[tensor product]] operation on [[finite abelian category|finite abelian categories]] is known as the \emph{Deligne tensor product} or \emph{Deligne box product}, introduced in (\hyperlink{Deligne}{Deligne 90}).

For $A$ and $B$ two [[abelian categories]], their Deligne tensor product $A \boxtimes B$ is the abelian category such that for any other abelian category $C$ right [[exact functors]] of the form $A \boxtimes B \to C$ are equivalent to functors $A \times B \to C$ that are right exact in each argument separately.

This tensor product exists for [[finite abelian categories]] but not generally on all [[abelian categories]]. However a slight variant does: instead of abelian categories one can consider categories with [[finite colimits]], see \hyperlink{Franco12}{Franco 12} and (\hyperlink{ChirvasituJohnson-Freyd11}{Chirvasitu\&Johnson-Freyd 11, remark 2.2.8}).

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

\hypertarget{in_terms_of_categories_of_modules_and_tensor_product_of_algebras}{}\subsubsection*{{In terms of categories of modules and tensor product of algebras}}\label{in_terms_of_categories_of_modules_and_tensor_product_of_algebras}

Recall that for every [[finite abelian category]] over $k$, there is a finite-dimensional [[associative algebra|algebra]] $A$ over $k$ and a $k$-linear [[equivalence of categories]]

\begin{displaymath}
\mathcal{C} \simeq A Mod_{fd}
  \,
\end{displaymath}
where the right side consists of $A$-modules which are finite-dimensional as vector spaces over $k$. $A$ is uniquely determined up to [[Morita equivalence]].

\begin{prop}
\label{}\hypertarget{}{}
For $A, B \in Alg_k$ two finite-dimensional [[associative algebras]] over a field $k$, the Deligne tensor product of their categories of finite-dimensional modules is the category of finite-dimensional modules of the [[tensor product of algebras]] $A \otimes_k B$:

\begin{displaymath}
A Mod_{fd} 
  \boxtimes
  B Mod_{fd}
  \simeq
  (A \otimes_k B) Mod_{fd}
  \,.
\end{displaymath}
\end{prop}
This appears for instance as (\hyperlink{EGNO}{EGNO, prop. 1.46.2}). Without the finiteness constraints and using the tensor product of categories with finite colimits, this appears as (\hyperlink{ChirvasituJohnson-Freyd11}{Chirvasitu\&Johnson-Freyd 11, remark 2.2.8}).

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

\begin{itemize}%
\item [[tensor product of presentable (infinity,1)-categories]]

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

The construction was introduced in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Cat\'e{}gories tannakiennes}, The Grothendieck Festschrift, Vol. II. Progr. Math. 87, 111--195. Birkh\"a{}user Boston. 1990 (1990)

\end{itemize}
A survey is in

\begin{itemize}%
\item [[Ignacio L\'o{}pez Franco]], \emph{Tensor products of finitely cococomplete and abelian categories} (2012) (\href{http://www.mat.uc.pt/~workCT/slides/Ignacio.pdf}{pdf})

\end{itemize}
Here the author points out that while Deligne's tensor product always exists for finite abelian categories, it does not always exist for general [[abelian categories]]. He argues that in this case it is better to use Kelly's tensor product of finitely cocomplete categories, because it always exists, and it agrees with Deligne's tensor product when the latter exists.

Kelly's tensor product can be found in section 6.5 of

\begin{itemize}%
\item [[Max Kelly]], \emph{Basic concepts of enriched category theory}, London Math. Soc. Lec. Note Series \textbf{64}, Cambridge Univ. Press 1982, 245 pp.. Reprint: TAC reprints 10, \href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html}{tac},\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf}{pdf}

\end{itemize}
Similar remarks (in the context of [[2-rings]]/[[2-modules]]) are from corollary 2.2.5 on in

\begin{itemize}%
\item Alexandru Chirvasitu, [[Theo Johnson-Freyd]], \emph{The fundamental pro-groupoid of an affine 2-scheme} (\href{http://arxiv.org/abs/1105.3104}{arXiv:1105.3104})

\end{itemize}
In this set of lecture notes

\begin{itemize}%
\item [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], \emph{Topics in Lie theory and Tensor categories}, Lecture notes (Spring 2009) (\href{http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/}{web})

\end{itemize}
the Deligne tensor product is discussed in \href{http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf}{lecture 9}.

[[!redirects tensor product of abelian categories]]

[[!redirects Deligne tensor product]] [[!redirects Deligne tensor products]]



\end{document}